Search | arXiv e-print repository

<!DOCTYPE html> <html lang="en"> <head> <meta charset="utf-8"/> <meta name="viewport" content="width=device-width, initial-scale=1"/>  <link rel="apple-touch-icon" sizes="180x180" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/apple-touch-icon.png"> <link rel="icon" type="image/png" sizes="32x32" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/favicon-32x32.png"> <link rel="icon" type="image/png" sizes="16x16" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/favicon-16x16.png"> <link rel="manifest" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/site.webmanifest"> <link rel="mask-icon" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/safari-pinned-tab.svg" color="#b31b1b"> <link rel="shortcut icon" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/favicon.ico"> <meta name="msapplication-TileColor" content="#b31b1b"> <meta name="msapplication-config" content="images/icons/browserconfig.xml"> <meta name="theme-color" content="#b31b1b">  <title>Search | arXiv e-print repository</title> <script defer src="https://static.arxiv.org/static/base/1.0.0a5/fontawesome-free-5.11.2-web/js/all.js"></script> <link rel="stylesheet" href="https://static.arxiv.org/static/base/1.0.0a5/css/arxivstyle.css" /> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ messageStyle: "none", extensions: ["tex2jax.js"], jax: ["input/TeX", "output/HTML-CSS"], tex2jax: { inlineMath: [ ['$','$'], ["\$","\$"] ], displayMath: [ ['$$','$$'], ["\\[","\\]"] ], processEscapes: true, ignoreClass: '.*', processClass: 'mathjax.*' }, TeX: { extensions: ["AMSmath.js", "AMSsymbols.js", "noErrors.js"], noErrors: { inlineDelimiters: ["$","$"], multiLine: false, style: { "font-size": "normal", "border": "" } } }, "HTML-CSS": { availableFonts: ["TeX"] } }); </script> <script src='//static.arxiv.org/MathJax-2.7.3/MathJax.js'></script> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/notification.js"></script> <link rel="stylesheet" href="https://static.arxiv.org/static/search/0.5.6/css/bulma-tooltip.min.css" /> <link rel="stylesheet" href="https://static.arxiv.org/static/search/0.5.6/css/search.css" /> <script src="https://code.jquery.com/jquery-3.2.1.slim.min.js" integrity="sha256-k2WSCIexGzOj3Euiig+TlR8gA0EmPjuc79OEeY5L45g=" crossorigin="anonymous"></script> <script src="https://static.arxiv.org/static/search/0.5.6/js/fieldset.js"></script> <style> radio#cf-customfield_11400 { display: none; } </style> </head> <body> <header><a href="#main-container" class="is-sr-only">Skip to main content</a>  <div class="attribution level is-marginless" role="banner"> <div class="level-left"> <a class="level-item" href="https://cornell.edu/"><img src="https://static.arxiv.org/static/base/1.0.0a5/images/cornell-reduced-white-SMALL.svg" alt="Cornell University" width="200" aria-label="logo" /></a> </div> <div class="level-right is-marginless"><p class="sponsors level-item is-marginless"><span id="support-ack-url">We gratefully acknowledge support from<br /> the Simons Foundation, <a href="https://info.arxiv.org/about/ourmembers.html">member institutions</a>, and all contributors. <a href="https://info.arxiv.org/about/donate.html">Donate</a></span></p></div> </div>  <div class="identity level is-marginless"> <div class="level-left"> <div class="level-item"> <a class="arxiv" href="https://arxiv.org/" aria-label="arxiv-logo"> <img src="https://static.arxiv.org/static/base/1.0.0a5/images/arxiv-logo-one-color-white.svg" aria-label="logo" alt="arxiv logo" width="85" style="width:85px;"/> </a> </div> </div> <div class="search-block level-right"> <form class="level-item mini-search" method="GET" action="https://arxiv.org/search"> <div class="field has-addons"> <div class="control"> <input class="input is-small" type="text" name="query" placeholder="Search..." aria-label="Search term or terms" /> <p class="help"><a href="https://info.arxiv.org/help">Help</a> | <a href="https://arxiv.org/search/advanced">Advanced Search</a></p> </div> <div class="control"> <div class="select is-small"> <select name="searchtype" aria-label="Field to search"> <option value="all" selected="selected">All fields</option> <option value="title">Title</option> <option value="author">Author</option> <option value="abstract">Abstract</option> <option value="comments">Comments</option> <option value="journal_ref">Journal reference</option> <option value="acm_class">ACM classification</option> <option value="msc_class">MSC classification</option> <option value="report_num">Report number</option> <option value="paper_id">arXiv identifier</option> <option value="doi">DOI</option> <option value="orcid">ORCID</option> <option value="author_id">arXiv author ID</option> <option value="help">Help pages</option> <option value="full_text">Full text</option> </select> </div> </div> <input type="hidden" name="source" value="header"> <button class="button is-small is-cul-darker">Search</button> </div> </form> </div> </div>  <div class="container"> <div class="user-tools is-size-7 has-text-right has-text-weight-bold" role="navigation" aria-label="User menu"> <a href="https://arxiv.org/login">Login</a> </div> </div> </header> <main class="container" id="main-container"> <div class="level is-marginless"> <div class="level-left"> <h1 class="title is-clearfix"> Showing 1–19 of 19 results for author: <span class="mathjax">Ozbudak, F</span> </h1> </div> <div class="level-right is-hidden-mobile">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> <div class="content"> <form method="GET" action="/search/cs" aria-role="search"> Searching in archive <strong>cs</strong>. <a href="/search/?searchtype=author&query=Ozbudak%2C+F">Search in all archives.</a> <div class="field has-addons-tablet"> <div class="control is-expanded"> <label for="query" class="hidden-label">Search term or terms</label> <input class="input is-medium" id="query" name="query" placeholder="Search term..." type="text" value="Ozbudak, F"> </div> <div class="select control is-medium"> <label class="is-hidden" for="searchtype">Field</label> <select class="is-medium" id="searchtype" name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> </div> <div class="control"> <button class="button is-link is-medium">Search</button> </div> </div> <div class="field"> <div class="control is-size-7"> <label class="radio"> <input checked id="abstracts-0" name="abstracts" type="radio" value="show"> Show abstracts </label> <label class="radio"> <input id="abstracts-1" name="abstracts" type="radio" value="hide"> Hide abstracts </label> </div> </div> <div class="is-clearfix" style="height: 2.5em"> <div class="is-pulled-right"> <a href="/search/advanced?terms-0-term=Ozbudak%2C+F&terms-0-field=author&size=50&order=-announced_date_first">Advanced Search</a> </div> </div> <input type="hidden" name="order" value="-announced_date_first"> <input type="hidden" name="size" value="50"> </form> <div class="level breathe-horizontal"> <div class="level-left"> <form method="GET" action="/search/"> <div style="display: none;"> <select id="searchtype" name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Ozbudak, F"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.16559">arXiv:2411.16559</a> <span> [<a href="https://arxiv.org/pdf/2411.16559">pdf</a>, <a href="https://arxiv.org/ps/2411.16559">ps</a>, <a href="https://arxiv.org/format/2411.16559">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> Generalizing the Bierbrauer-Friedman bound </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Krotov%2C+D+S">Denis S. Krotov</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a>, <a href="/search/cs?searchtype=author&query=Potapov%2C+V+N">Vladimir N. Potapov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.16559v1-abstract-short" style="display: inline;"> We characterize mixed-level orthogonal arrays it terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16559v1-abstract-full').style.display = 'inline'; document.getElementById('2411.16559v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.16559v1-abstract-full" style="display: none;"> We characterize mixed-level orthogonal arrays it terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic $t$-design, mixed orthogonal array, completely-regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16559v1-abstract-full').style.display = 'none'; document.getElementById('2411.16559v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05B15; 05E30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14087">arXiv:2411.14087</a> <span> [<a href="https://arxiv.org/pdf/2411.14087">pdf</a>, <a href="https://arxiv.org/ps/2411.14087">ps</a>, <a href="https://arxiv.org/format/2411.14087">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Determining the covering radius of all generalized Zetterberg codes in odd characteristic </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Shi%2C+M">Minjia Shi</a>, <a href="/search/cs?searchtype=author&query=Li%2C+S">Shitao Li</a>, <a href="/search/cs?searchtype=author&query=Helleseth%2C+T">Tor Helleseth</a>, <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14087v1-abstract-short" style="display: inline;"> For an integer $s\ge 1$, let $\mathcal{C}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and 脰zbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, and left the remaining case as an open problem. In this pape… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14087v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14087v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14087v1-abstract-full" style="display: none;"> For an integer $s\ge 1$, let $\mathcal{C}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and 脰zbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for $q_0^s \equiv 7 \pmod{8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length $\frac{q_0^s+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14087v1-abstract-full').style.display = 'none'; document.getElementById('2411.14087v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.18996">arXiv:2405.18996</a> <span> [<a href="https://arxiv.org/pdf/2405.18996">pdf</a>, <a href="https://arxiv.org/ps/2405.18996">ps</a>, <a href="https://arxiv.org/format/2405.18996">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Using multi-orbit cyclic subspace codes for constructing optical orthogonal codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a>, <a href="/search/cs?searchtype=author&query=Santonastaso%2C+P">Paolo Santonastaso</a>, <a href="/search/cs?searchtype=author&query=Zullo%2C+F">Ferdinando Zullo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.18996v1-abstract-short" style="display: inline;"> We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches using combinatorial and additive (character sum) structures of finite fields. Consequently, we immediately obtain new classes of optical orthogonal codes with diffe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.18996v1-abstract-full').style.display = 'inline'; document.getElementById('2405.18996v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.18996v1-abstract-full" style="display: none;"> We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches using combinatorial and additive (character sum) structures of finite fields. Consequently, we immediately obtain new classes of optical orthogonal codes with different parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.18996v1-abstract-full').style.display = 'none'; document.getElementById('2405.18996v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.15057">arXiv:2405.15057</a> <span> [<a href="https://arxiv.org/pdf/2405.15057">pdf</a>, <a href="https://arxiv.org/format/2405.15057">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Ezerman%2C+M+F">Martianus Frederic Ezerman</a>, <a href="/search/cs?searchtype=author&query=Grassl%2C+M">Markus Grassl</a>, <a href="/search/cs?searchtype=author&query=Ling%2C+S">San Ling</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a>, <a href="/search/cs?searchtype=author&query=%C3%96zkaya%2C+B">Buket 脰zkaya</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.15057v2-abstract-short" style="display: inline;"> Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15057v2-abstract-full').style.display = 'inline'; document.getElementById('2405.15057v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.15057v2-abstract-full" style="display: none;"> Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15057v2-abstract-full').style.display = 'none'; document.getElementById('2405.15057v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 8 tables; see also http://codetables.de This work has been submitted to the IEEE for possible publication; v2: corrected some typos, considerably expanded the tables with new quantum codes</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.04941">arXiv:2401.04941</a> <span> [<a href="https://arxiv.org/pdf/2401.04941">pdf</a>, <a href="https://arxiv.org/ps/2401.04941">ps</a>, <a href="https://arxiv.org/format/2401.04941">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Griesmer Bound and Constructions of Linear Codes in $b$-Symbol Metric </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Luo%2C+G">Gaojun Luo</a>, <a href="/search/cs?searchtype=author&query=Ezerman%2C+M+F">Martianus Frederic Ezerman</a>, <a href="/search/cs?searchtype=author&query=G%C3%BCneri%2C+C">Cem G眉neri</a>, <a href="/search/cs?searchtype=author&query=Ling%2C+S">San Ling</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.04941v1-abstract-short" style="display: inline;"> The $b$-symbol metric is a generalization of the Hamming metric. Linear codes, in the $b$-symbol metric, have been used in the read channel whose outputs consist of $b$ consecutive symbols. The Griesmer bound outperforms the Singleton bound for $\mathbb{F}_q$-linear codes in the Hamming metric, when $q$ is fixed and the length is large enough. This scenario is also applicable in the $b$-symbol met… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.04941v1-abstract-full').style.display = 'inline'; document.getElementById('2401.04941v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.04941v1-abstract-full" style="display: none;"> The $b$-symbol metric is a generalization of the Hamming metric. Linear codes, in the $b$-symbol metric, have been used in the read channel whose outputs consist of $b$ consecutive symbols. The Griesmer bound outperforms the Singleton bound for $\mathbb{F}_q$-linear codes in the Hamming metric, when $q$ is fixed and the length is large enough. This scenario is also applicable in the $b$-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the $b$-symbol metric. In this paper, we present the $b$-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the $b$-symbol Griesmer bound. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.04941v1-abstract-full').style.display = 'none'; document.getElementById('2401.04941v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.00354">arXiv:2311.00354</a> <span> [<a href="https://arxiv.org/pdf/2311.00354">pdf</a>, <a href="https://arxiv.org/ps/2311.00354">ps</a>, <a href="https://arxiv.org/format/2311.00354">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> </div> </div> <p class="title is-5 mathjax"> Butson Hadamard matrices, bent sequences, and spherical codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Shi%2C+M">Minjia Shi</a>, <a href="/search/cs?searchtype=author&query=Lu%2C+D">Danni Lu</a>, <a href="/search/cs?searchtype=author&query=Armario%2C+A">Andr茅s Armario</a>, <a href="/search/cs?searchtype=author&query=Egan%2C+R">Ronan Egan</a>, <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a>, <a href="/search/cs?searchtype=author&query=Sol%C3%A9%2C+P">Patrick Sol茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.00354v1-abstract-short" style="display: inline;"> We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity, an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic field of order $q.$ In particular we construct self-dual bent sequences for various $q\le 60$ and lengths $n\le 21.$ Computational construction methods comprise the resol… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00354v1-abstract-full').style.display = 'inline'; document.getElementById('2311.00354v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.00354v1-abstract-full" style="display: none;"> We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity, an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic field of order $q.$ In particular we construct self-dual bent sequences for various $q\le 60$ and lengths $n\le 21.$ Computational construction methods comprise the resolution of polynomial systems by Groebner bases and eigenspace computations. Infinite families can be constructed from regular Hadamard matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions.As an application, we estimate the covering radius of the code attached to that matrix over $\Z_q.$ We derive a lower bound on that quantity for the Chinese Euclidean metric when bent sequences exist. We give the Euclidean distance spectrum, and bound above the covering radius of an attached spherical code, depending on its strength as a spherical design. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00354v1-abstract-full').style.display = 'none'; document.getElementById('2311.00354v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.02735">arXiv:2305.02735</a> <span> [<a href="https://arxiv.org/pdf/2305.02735">pdf</a>, <a href="https://arxiv.org/ps/2305.02735">ps</a>, <a href="https://arxiv.org/format/2305.02735">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1109/TIT.2023.3272566">10.1109/TIT.2023.3272566 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Shi%2C+M">Minjia Shi</a>, <a href="/search/cs?searchtype=author&query=Li%2C+X">Xiaoxiao Li</a>, <a href="/search/cs?searchtype=author&query=Krotov%2C+D+S">Denis S. Krotov</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.02735v1-abstract-short" style="display: inline;"> The Galois ring GR$(4^螖)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $螖$ over $Z_4$. For any odd $螖$ larger than $1$, we construct a partition of GR$(4^螖) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.02735v1-abstract-full').style.display = 'inline'; document.getElementById('2305.02735v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.02735v1-abstract-full" style="display: none;"> The Galois ring GR$(4^螖)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $螖$ over $Z_4$. For any odd $螖$ larger than $1$, we construct a partition of GR$(4^螖) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^螖)$ and, if $螖$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^螖)$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^螖-1)$ in $D((2^螖-1)(2^螖-2)/{6}, 2^螖-1 )$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.02735v1-abstract-full').style.display = 'none'; document.getElementById('2305.02735v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted version; 7 IEEE TIT pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B99 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> IEEE Trans. Inf. Theory 69(9) 2023, 5597-5603 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.00298">arXiv:2211.00298</a> <span> [<a href="https://arxiv.org/pdf/2211.00298">pdf</a>, <a href="https://arxiv.org/ps/2211.00298">ps</a>, <a href="https://arxiv.org/format/2211.00298">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1002/jcd.21931">10.1002/jcd.21931 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Constructing MRD codes by switching </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Shi%2C+M">Minjia Shi</a>, <a href="/search/cs?searchtype=author&query=Krotov%2C+D+S">Denis S. Krotov</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2211.00298v1-abstract-short" style="display: inline;"> MRD codes are maximum codes in the rank-distance metric space on $m$-by-$n$ matrices over the finite field of order $q$. They are diameter perfect and have the cardinality $q^{m(n-d+1)}$ if $m\ge n$. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twis… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.00298v1-abstract-full').style.display = 'inline'; document.getElementById('2211.00298v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.00298v1-abstract-full" style="display: none;"> MRD codes are maximum codes in the rank-distance metric space on $m$-by-$n$ matrices over the finite field of order $q$. They are diameter perfect and have the cardinality $q^{m(n-d+1)}$ if $m\ge n$. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $m$ if the other parameters ($n$, $q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. Keywords: MRD codes, rank distance, bilinear forms graph, switching, diameter perfect codes <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.00298v1-abstract-full').style.display = 'none'; document.getElementById('2211.00298v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B25 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Comb. Des. 32(5) 2024, 219-237 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.00805">arXiv:2110.00805</a> <span> [<a href="https://arxiv.org/pdf/2110.00805">pdf</a>, <a href="https://arxiv.org/ps/2110.00805">ps</a>, <a href="https://arxiv.org/format/2110.00805">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Complete b-symbol weight distribution of some irreducible cyclic codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Zhu%2C+H">Hongwei Zhu</a>, <a href="/search/cs?searchtype=author&query=Shi%2C+M">Minjia Shi</a>, <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.00805v1-abstract-short" style="display: inline;"> Recently, $b$-symbol codes are proposed to protect against $b$-symbol errors in $b$-symbol read channels. It is an interesting subject of study to consider the complete $b$-symbol weight distribution of cyclic codes since $b$-symbol metric is a generalization for Hamming metric. The complete $b$-symbol Hamming weight distribution of irreducible codes is known in only a few cases. In this paper, we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.00805v1-abstract-full').style.display = 'inline'; document.getElementById('2110.00805v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.00805v1-abstract-full" style="display: none;"> Recently, $b$-symbol codes are proposed to protect against $b$-symbol errors in $b$-symbol read channels. It is an interesting subject of study to consider the complete $b$-symbol weight distribution of cyclic codes since $b$-symbol metric is a generalization for Hamming metric. The complete $b$-symbol Hamming weight distribution of irreducible codes is known in only a few cases. In this paper, we give a complete $b$-symbol Hamming weight distribution of a class of irreducible codes with two nonzero $b$-symbol Hamming weights. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.00805v1-abstract-full').style.display = 'none'; document.getElementById('2110.00805v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.04407">arXiv:2103.04407</a> <span> [<a href="https://arxiv.org/pdf/2103.04407">pdf</a>, <a href="https://arxiv.org/ps/2103.04407">ps</a>, <a href="https://arxiv.org/format/2103.04407">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> LCD Codes from tridiagonal Toeplitz matrice </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Shi%2C+M">Minjia Shi</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a>, <a href="/search/cs?searchtype=author&query=Xu%2C+L">Li Xu</a>, <a href="/search/cs?searchtype=author&query=Sol%C3%A9%2C+P">Patrick Sol茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.04407v1-abstract-short" style="display: inline;"> Double Toeplitz (DT) codes are codes with a generator matrix of the form $(I,T)$ with $T$ a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When $T$ is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or qua… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.04407v1-abstract-full').style.display = 'inline'; document.getElementById('2103.04407v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.04407v1-abstract-full" style="display: none;"> Double Toeplitz (DT) codes are codes with a generator matrix of the form $(I,T)$ with $T$ a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When $T$ is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.04407v1-abstract-full').style.display = 'none'; document.getElementById('2103.04407v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05; 15B05; 12E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.06081">arXiv:1909.06081</a> <span> [<a href="https://arxiv.org/pdf/1909.06081">pdf</a>, <a href="https://arxiv.org/ps/1909.06081">ps</a>, <a href="https://arxiv.org/format/1909.06081">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10623-020-00732-z">10.1007/s10623-020-00732-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Subspace Packings -- Constructions and Bounds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Etzion%2C+T">Tuvi Etzion</a>, <a href="/search/cs?searchtype=author&query=Kurz%2C+S">Sascha Kurz</a>, <a href="/search/cs?searchtype=author&query=Otal%2C+K">Kamil Otal</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.06081v2-abstract-short" style="display: inline;"> The Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. K枚tter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e., constant dimension codes. These codes of the Grassmannian… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.06081v2-abstract-full').style.display = 'inline'; document.getElementById('1909.06081v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.06081v2-abstract-full" style="display: none;"> The Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. K枚tter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e., constant dimension codes. These codes of the Grassmannian $\mathcal{G}_q(n,k)$ also form a family of $q$-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of $q$-analogs of block designs which was not considered before. This family, called subspace packings, is the $q$-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing $t$-$(n,k,位)_q$ is a set $\mathcal{S}$ of $k$-subspaces from $\mathcal{G}_q(n,k)$ such that each $t$-subspace of $\mathcal{G}_q(n,t)$ is contained in at most $位$ elements of $\mathcal{S}$. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.06081v2-abstract-full').style.display = 'none'; document.getElementById('1909.06081v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, 27 tables, continuation of arXiv:1811.04611, typos corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B65; 94B60; 51E20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.04611">arXiv:1811.04611</a> <span> [<a href="https://arxiv.org/pdf/1811.04611">pdf</a>, <a href="https://arxiv.org/ps/1811.04611">ps</a>, <a href="https://arxiv.org/format/1811.04611">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Subspace Packings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Etzion%2C+T">Tuvi Etzion</a>, <a href="/search/cs?searchtype=author&query=Kurz%2C+S">Sascha Kurz</a>, <a href="/search/cs?searchtype=author&query=Otal%2C+K">Kamil Otal</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.04611v2-abstract-short" style="display: inline;"> The Grassmannian ${\mathcal G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. It is well known that codes in the Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian ${\mathcal G}_q(n,k)$ al… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.04611v2-abstract-full').style.display = 'inline'; document.getElementById('1811.04611v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.04611v2-abstract-full" style="display: none;"> The Grassmannian ${\mathcal G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. It is well known that codes in the Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian ${\mathcal G}_q(n,k)$ also form a family of $q$-analogs of block designs and they are called \emph{subspace designs}. The application of subspace codes has motivated extensive work on the $q$-analogs of block designs. In this paper, we examine one of the last families of $q$-analogs of block designs which was not considered before. This family called \emph{subspace packings} is the $q$-analog of packings. This family of designs was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A \emph{subspace packing} $t$-$(n,k,位)^m_q$ is a set $\mathcal{S}$ of $k$-subspaces from ${\mathcal G}_q(n,k)$ such that each $t$-subspace of ${\mathcal G}_q(n,t)$ is contained in at most $位$ elements of $\mathcal{S}$. The goal of this work is to consider the largest size of such subspace packings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.04611v2-abstract-full').style.display = 'none'; document.getElementById('1811.04611v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, 3 tables, typos corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05B40; 51E20; 11T71; 94B25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.07631">arXiv:1706.07631</a> <span> [<a href="https://arxiv.org/pdf/1706.07631">pdf</a>, <a href="https://arxiv.org/ps/1706.07631">ps</a>, <a href="https://arxiv.org/format/1706.07631">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> New cubic self-dual codes of length 54, 60 and 66 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=%C3%87omak%2C+P">P谋nar 脟omak</a>, <a href="/search/cs?searchtype=author&query=Kim%2C+J">Jon-Lark Kim</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1706.07631v1-abstract-short" style="display: inline;"> We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.07631v1-abstract-full').style.display = 'inline'; document.getElementById('1706.07631v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.07631v1-abstract-full" style="display: none;"> We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.07631v1-abstract-full').style.display = 'none'; document.getElementById('1706.07631v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11T71 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.05688">arXiv:1706.05688</a> <span> [<a href="https://arxiv.org/pdf/1706.05688">pdf</a>, <a href="https://arxiv.org/ps/1706.05688">ps</a>, <a href="https://arxiv.org/format/1706.05688">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> On affine variety codes from the Klein quartic </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Geil%2C+O">Olav Geil</a>, <a href="/search/cs?searchtype=author&query=%C3%94zbudak%2C+F">Ferruh 脭zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1706.05688v1-abstract-short" style="display: inline;"> We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Ex. 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [10, 7]… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.05688v1-abstract-full').style.display = 'inline'; document.getElementById('1706.05688v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.05688v1-abstract-full" style="display: none;"> We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Ex. 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [10, 7] from Gr枚bner basis theory and for this purpose we develop a new method where we inspired by Buchberger's algorithm perform a series of symbolic computations. 1 <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.05688v1-abstract-full').style.display = 'none'; document.getElementById('1706.05688v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11G50; 11T71; 94B65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1703.08362">arXiv:1703.08362</a> <span> [<a href="https://arxiv.org/pdf/1703.08362">pdf</a>, <a href="https://arxiv.org/ps/1703.08362">ps</a>, <a href="https://arxiv.org/format/1703.08362">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> A new class of three-weight linear codes from weakly regular plateaued functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Mesnager%2C+S">Sihem Mesnager</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a>, <a href="/search/cs?searchtype=author&query=S%C4%B1nak%2C+A">Ahmet S谋nak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1703.08362v1-abstract-short" style="display: inline;"> Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary line… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.08362v1-abstract-full').style.display = 'inline'; document.getElementById('1703.08362v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1703.08362v1-abstract-full" style="display: none;"> Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.08362v1-abstract-full').style.display = 'none'; document.getElementById('1703.08362v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 March, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The Extended Abstract of this work was submitted to WCC-2017 (the Tenth International Workshop on Coding and Cryptography)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.06672">arXiv:1701.06672</a> <span> [<a href="https://arxiv.org/pdf/1701.06672">pdf</a>, <a href="https://arxiv.org/ps/1701.06672">ps</a>, <a href="https://arxiv.org/format/1701.06672">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Additive cyclic codes over finite commutative chain rings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Mart%C3%ADnez-Moro%2C+E">Edgar Mart铆nez-Moro</a>, <a href="/search/cs?searchtype=author&query=Otal%2C+K">Kamil Otal</a>, <a href="/search/cs?searchtype=author&query=%C3%96zbudak%2C+F">Ferruh 脰zbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1701.06672v1-abstract-short" style="display: inline;"> Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the previous studies, whereas the other one has some unusu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.06672v1-abstract-full').style.display = 'inline'; document.getElementById('1701.06672v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.06672v1-abstract-full" style="display: none;"> Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the previous studies, whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.06672v1-abstract-full').style.display = 'none'; document.getElementById('1701.06672v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11T71; 94B99; 81P70; 13M10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1210.0140">arXiv:1210.0140</a> <span> [<a href="https://arxiv.org/pdf/1210.0140">pdf</a>, <a href="https://arxiv.org/ps/1210.0140">ps</a>, <a href="https://arxiv.org/format/1210.0140">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Lopez-Permouth%2C+S+R">Sergio R. Lopez-Permouth</a>, <a href="/search/cs?searchtype=author&query=Ozadam%2C+H">Hakan Ozadam</a>, <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a>, <a href="/search/cs?searchtype=author&query=Szabo%2C+S">Steve Szabo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1210.0140v1-abstract-short" style="display: inline;"> Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.0140v1-abstract-full').style.display = 'inline'; document.getElementById('1210.0140v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1210.0140v1-abstract-full" style="display: none;"> Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.0140v1-abstract-full').style.display = 'none'; document.getElementById('1210.0140v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 September, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax"> arXiv admin note: text overlap with arXiv:0906.4008</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1003.3386">arXiv:1003.3386</a> <span> [<a href="https://arxiv.org/pdf/1003.3386">pdf</a>, <a href="https://arxiv.org/ps/1003.3386">ps</a>, <a href="https://arxiv.org/format/1003.3386">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Monomial-like codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Martinez-Moro%2C+E">Edgar Martinez-Moro</a>, <a href="/search/cs?searchtype=author&query=Ozadam%2C+H">Hakan Ozadam</a>, <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a>, <a href="/search/cs?searchtype=author&query=Szabo%2C+S">Steve Szabo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1003.3386v1-abstract-short" style="display: inline;"> As a generalization of cyclic codes of length p^s over F_{p^a}, we study n-dimensional cyclic codes of length p^{s_1} X ... X p^{s_n} over F_{p^a} generated by a single "monomial". Namely, we study multi-variable cyclic codes of the form <(x_1 - 1)^{i_1} ... (x_n - 1)^{i_n}> in F_{p^a}[x_1...x_n] / < x_1^{p^{s_1}}-1, ..., x_n^{p^{s_n}}-1 >. We call such codes monomial-like codes. We show that the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1003.3386v1-abstract-full').style.display = 'inline'; document.getElementById('1003.3386v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1003.3386v1-abstract-full" style="display: none;"> As a generalization of cyclic codes of length p^s over F_{p^a}, we study n-dimensional cyclic codes of length p^{s_1} X ... X p^{s_n} over F_{p^a} generated by a single "monomial". Namely, we study multi-variable cyclic codes of the form <(x_1 - 1)^{i_1} ... (x_n - 1)^{i_n}> in F_{p^a}[x_1...x_n] / < x_1^{p^{s_1}}-1, ..., x_n^{p^{s_n}}-1 >. We call such codes monomial-like codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. We determine the dual of monomial-like codes yielding a parity check matrix. We also present an alternative way of constructing a parity check matrix using the Hasse derivative. We study the weight hierarchy of certain monomial like codes. We simplify an expression that gives us the weight hierarchy of these codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1003.3386v1-abstract-full').style.display = 'none'; document.getElementById('1003.3386v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2010. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0906.4008">arXiv:0906.4008</a> <span> [<a href="https://arxiv.org/pdf/0906.4008">pdf</a>, <a href="https://arxiv.org/ps/0906.4008">ps</a>, <a href="https://arxiv.org/format/0906.4008">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Two generalizations on the minimum Hamming distance of repeated-root constacyclic codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Ozadam%2C+H">Hakan Ozadam</a>, <a href="/search/cs?searchtype=author&query=Ozbudak%2C+F">Ferruh Ozbudak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0906.4008v1-abstract-short" style="display: inline;"> We study constacyclic codes, of length $np^s$ and $2np^s$, that are generated by the polynomials $(x^n + 纬)^{\ell}$ and $(x^n - 尉)^i(x^n + 尉)^j$\ respectively, where $x^n + 纬$, $x^n - 尉$ and $x^n + 尉$ are irreducible over the alphabet $\F_{p^a}$. We generalize the results of [5], [6] and [7] by computing the minimum Hamming distance of these codes. As a particular case, we determine the minimum… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0906.4008v1-abstract-full').style.display = 'inline'; document.getElementById('0906.4008v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0906.4008v1-abstract-full" style="display: none;"> We study constacyclic codes, of length $np^s$ and $2np^s$, that are generated by the polynomials $(x^n + 纬)^{\ell}$ and $(x^n - 尉)^i(x^n + 尉)^j$\ respectively, where $x^n + 纬$, $x^n - 尉$ and $x^n + 尉$ are irreducible over the alphabet $\F_{p^a}$. We generalize the results of [5], [6] and [7] by computing the minimum Hamming distance of these codes. As a particular case, we determine the minimum Hamming distance of cyclic and negacyclic codes, of length $2p^s$, over a finite field of characteristic $p$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0906.4008v1-abstract-full').style.display = 'none'; document.getElementById('0906.4008v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 June, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">We do not plan to publish the results of this paper on their own. We have put this paper for referring purposes</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

CINXE.COM

Search | arXiv e-print repository