Document Zbl 1508.94053

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(2022)</a>. </div> <div class="abstract">Summary: In this paper, let $\mathbb{F}_q$ be a finite field with $q$ elements and $m_1, m_2$ two positive integers with $d = \gcd(m_1, m_2)$. Let $\mathfrak{s}^{(1)} = (\operatorname{T}_1 (\alpha_1^{e i}))_{i = 0}^{n - 1}$ and $\mathfrak{s}^{(2)} = (\operatorname{T}_2 (\alpha_2^i))_{i = 0}^{n - 1}$ be two $q$-ary sequences of period $q^{m_1} - 1$ and $q^{m_2} - 1$, respectively, where $\mathbb{F}_{q^{m_i}}^\ast = \langle \alpha_i \rangle, \operatorname{T}_i$ denote the trace function from $\mathbb{F}_{q^{m_i}}$ to $\mathbb{F}_q$, $i = 1, 2$, $n = (q^{m_1} - 1)( q^{m_2} - 1) /( q^d - 1)$, and $\gcd(e, q^{m_1} - 1) = 1$.<br class="zbmathjax-paragraph">If $e \equiv 1 \pmod{q^d - 1}$ and $m_2 = m_1 + d$, then two sequences have $p$ values cross correlations and are asymptotically optimal with respect to the cross correlation bound. If $e \equiv 1\pmod{q^d - 1}$ and $m_2 = m_1 + 2 d$, then the cross correlations have at most $q^d - 1$ values by closed-form expressions of Kloosterman sums over finite fields. If $e \equiv - 1 \pmod{q^d - 1}$, then the cross correlations also have at most $q^d - 1$ values. Furthermore, we support some examples by Magma program.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A55" title="MSC2020">94A55</a> </td> <td class="space"> Shift register sequences and sequences over finite alphabets in information and communication theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11T23" title="MSC2020">11T23</a> </td> <td class="space"> Exponential sums </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Across+correlation">cross correlation</a>; <a href="/?q=ut%3Aexponential+sums">exponential sums</a>; <a href="/?q=ut%3Afinite+field">finite field</a></div> <div class="software"> <h3>Software:</h3><a href="/software/540">Magma</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1508.94053" data-ciurl="/ci/07576663" data-biburl="/bibtex/07576663.bib" data-amsurl="/amsrefs/07576663.bib" data-xmlurl="/xml/07576663.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07576663.pdf" title="Zbl 1508.94053 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.ffa.2022.102077" aria-label="DOI for “The cross-correlation values of a class of sequences with different lengths”" title="10.1016/j.ffa.2022.102077">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Cusick, T. 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This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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