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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/2411.13375">arXiv:2411.13375</a> <span> [<a href="https://arxiv.org/pdf/2411.13375">pdf</a>, <a href="https://arxiv.org/format/2411.13375">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="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The weight hierarchy of decreasing norm-trace codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos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="2411.13375v1-abstract-short" style="display: inline;"> The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, $t$-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have bee… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13375v1-abstract-full').style.display = 'inline'; document.getElementById('2411.13375v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13375v1-abstract-full" style="display: none;"> The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, $t$-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by $x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$ over the finite field of cardinality $q^s$, where $u$ is a positive divisor of $\frac{q^s - 1}{q - 1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13375v1-abstract-full').style.display = 'none'; document.getElementById('2411.13375v1-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> 20 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> 94B05; 11T71; 14G50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.02916">arXiv:2408.02916</a> <span> [<a href="https://arxiv.org/pdf/2408.02916">pdf</a>, <a href="https://arxiv.org/ps/2408.02916">ps</a>, <a href="https://arxiv.org/format/2408.02916">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"> Binary Triorthogonal and CSS-T Codes for Quantum Error Correction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a>, <a href="/search/cs?searchtype=author&query=Soprunov%2C+I">Ivan Soprunov</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="2408.02916v1-abstract-short" style="display: inline;"> In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a cons… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02916v1-abstract-full').style.display = 'inline'; document.getElementById('2408.02916v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.02916v1-abstract-full" style="display: none;"> In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a consequence, we show that a binary triorthogonal matrix uniquely determines the parameters of the corresponding triorthogonal quantum code, meaning that any other equivalent matrix that is also triorthogonal gives rise to a triorthogonal quantum code with the same parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02916v1-abstract-full').style.display = 'none'; document.getElementById('2408.02916v1-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> 5 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.00425">arXiv:2406.00425</a> <span> [<a href="https://arxiv.org/pdf/2406.00425">pdf</a>, <a href="https://arxiv.org/ps/2406.00425">ps</a>, <a href="https://arxiv.org/format/2406.00425">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"> Toward Quantum CSS-T Codes from Sparse Matrices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=McMillon%2C+E">Emily McMillon</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="2406.00425v1-abstract-short" style="display: inline;"> CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair $(C_1, C_2)$ of binary linear codes $C_1$ and $C_2$ that satisfy certain conditions. We prove that $C_1$ and $C_2$ form a CSS-T pair if and only if $C_2 \subset \operatorname{Hull}(C_1) \cap \operatorname{Hull}(C_1^2)$, where the hull of a code is the intersection… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00425v1-abstract-full').style.display = 'inline'; document.getElementById('2406.00425v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.00425v1-abstract-full" style="display: none;"> CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair $(C_1, C_2)$ of binary linear codes $C_1$ and $C_2$ that satisfy certain conditions. We prove that $C_1$ and $C_2$ form a CSS-T pair if and only if $C_2 \subset \operatorname{Hull}(C_1) \cap \operatorname{Hull}(C_1^2)$, where the hull of a code is the intersection of the code with its dual. We show that if $(C_1,C_2)$ is a CSS-T pair, and the code $C_2$ is degenerated on $\{i\}$, meaning that the $i^{th}$-entry is zero for all the elements in $C_2$, then the pair of punctured codes $(C_1|_i,C_2|_i)$ is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00425v1-abstract-full').style.display = 'none'; document.getElementById('2406.00425v1-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.07201">arXiv:2404.07201</a> <span> [<a href="https://arxiv.org/pdf/2404.07201">pdf</a>, <a href="https://arxiv.org/ps/2404.07201">ps</a>, <a href="https://arxiv.org/format/2404.07201">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="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Fractional decoding of algebraic geometry codes over extension fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=Santos%2C+W">Welington Santos</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="2404.07201v1-abstract-short" style="display: inline;"> In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}_{q^\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07201v1-abstract-full').style.display = 'inline'; document.getElementById('2404.07201v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.07201v1-abstract-full" style="display: none;"> In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}_{q^\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07201v1-abstract-full').style.display = 'none'; document.getElementById('2404.07201v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.05445">arXiv:2403.05445</a> <span> [<a href="https://arxiv.org/pdf/2403.05445">pdf</a>, <a href="https://arxiv.org/ps/2403.05445">ps</a>, <a href="https://arxiv.org/format/2403.05445">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The minimum distance of a parameterized code over an even cycle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=Neves%2C+J">Jorge Neves</a>, <a href="/search/cs?searchtype=author&query=Sarmiento%2C+E">Eliseo Sarmiento</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="2403.05445v2-abstract-short" style="display: inline;"> We compute the minimum distance of the parameterized code of order 1 over an even cycle. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.05445v2-abstract-full" style="display: none;"> We compute the minimum distance of the parameterized code of order 1 over an even cycle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.05445v2-abstract-full').style.display = 'none'; document.getElementById('2403.05445v2-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.17518">arXiv:2312.17518</a> <span> [<a href="https://arxiv.org/pdf/2312.17518">pdf</a>, <a href="https://arxiv.org/ps/2312.17518">ps</a>, <a href="https://arxiv.org/format/2312.17518">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 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/s11128-024-04427-5">10.1007/s11128-024-04427-5 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a>, <a href="/search/cs?searchtype=author&query=Soprunov%2C+I">Ivan Soprunov</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="2312.17518v2-abstract-short" style="display: inline;"> CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17518v2-abstract-full').style.display = 'inline'; document.getElementById('2312.17518v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.17518v2-abstract-full" style="display: none;"> CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $(C_1, C_2)$ is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed-Muller, cyclic, and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17518v2-abstract-full').style.display = 'none'; document.getElementById('2312.17518v2-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05 (Primary); 81P70; 11T71; 14G50 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Quantum Inf Process 23, 230 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.14521">arXiv:2212.14521</a> <span> [<a href="https://arxiv.org/pdf/2212.14521">pdf</a>, <a href="https://arxiv.org/ps/2212.14521">ps</a>, <a href="https://arxiv.org/format/2212.14521">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"> Relative hulls and quantum codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Anderson%2C+S+E">Sarah E. Anderson</a>, <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=Soprunov%2C+I">Ivan Soprunov</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="2212.14521v2-abstract-short" style="display: inline;"> Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the $e$-Gal… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.14521v2-abstract-full').style.display = 'inline'; document.getElementById('2212.14521v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.14521v2-abstract-full" style="display: none;"> Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the $e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when $q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.14521v2-abstract-full').style.display = 'none'; document.getElementById('2212.14521v2-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 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05; 81P70; 11T71; 14G50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.12594">arXiv:2107.12594</a> <span> [<a href="https://arxiv.org/pdf/2107.12594">pdf</a>, <a href="https://arxiv.org/format/2107.12594">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="Commutative Algebra">math.AC</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/10.1142/S0219498825500628">10.1142/S0219498825500628 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the generalized Hamming weights of hyperbolic codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=Garc%C3%ADa-Marco%2C+I">Ignacio Garc铆a-Marco</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=M%C3%A1rquez-Corbella%2C+I">Irene M谩rquez-Corbella</a>, <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=Sarmiento%2C+E">Eliseo Sarmiento</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="2107.12594v2-abstract-short" style="display: inline;"> A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.12594v2-abstract-full').style.display = 'inline'; document.getElementById('2107.12594v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.12594v2-abstract-full" style="display: none;"> A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code and the smallest Reed-Muller in the hyperbolic code. We then prove that similarly to Reed-Muller and Cartesian codes, the $r$-th generalized Hamming weight and the $r$-th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the $r$-th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$-th footprint of a hyperbolic code that, sometimes, are sharp. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.12594v2-abstract-full').style.display = 'none'; document.getElementById('2107.12594v2-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> 30 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05; 13P25; 14G50; 11T71 </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> 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