Document Zbl 1392.94943

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Suppose that $\alpha_1$ and $\alpha_2$ are two primitive elements of $\mathbb{F}_{q^{m_1}}$ and $\mathbb{F}_{q^{m_2}}$, respectively. Let $n = (q^{m_1} - 1)(q^{m_2} - 1) /(q^d - 1)$ and $\text{T}_i$ denote the trace function from $\mathbb{F}_{q^{m_i}}$ to $\mathbb{F}_q$ for $i = 1, 2$. We define a cyclic code $\mathcal{C}_{(q, m_1, m_2)} = \{c(a, b) : a \in \mathbb{F}_{q^{m_1}},\;b \in \mathbb{F}_{q^{m_2}} \},$ where \[ c(a, b) = (\text{T}_1(a \alpha_1^0) + \text{T}_2(b \alpha_2^0), \text{T}_1(a \alpha_1^1) + \text{T}_2(b \alpha_2^1), \ldots, \text{T}_1(a \alpha_1^{n - 1}) + \text{T}_2(b \alpha_2^{n - 1})). \] In this paper, we use Gauss sums to investigate the weight distribution of $\mathcal{C}_{(q, m_1, m_2)}$ and prove that it has at most four nonzero weights if $d = 2$ and $\gcd (\frac{m_1 - m_2}{2}, q - 1) = 1$. Furthermore, we get a class of three-weight cyclic codes if $| m_1 - m_2 | = 2$. Some optimal or nearly optimal cyclic codes are presented.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6444829">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A94B15" title="MSC2020">94B15</a> </td> <td class="space"> Cyclic codes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11T71" title="MSC2020">11T71</a> </td> <td class="space"> Algebraic coding theory; cryptography (number-theoretic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11T24" title="MSC2020">11T24</a> </td> <td class="space"> Other character sums and Gauss sums </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aweight+distribution">weight distribution</a>; <a href="/?q=ut%3Acyclic+codes">cyclic codes</a>; <a href="/?q=ut%3AGauss+sums">Gauss sums</a></div> <div class="software"> <h3>Software:</h3><a href="/software/28701">Code Tables</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 1392.94943" data-ciurl="/ci/06444829" data-biburl="/bibtex/06444829.bib" data-amsurl="/amsrefs/06444829.bib" data-xmlurl="/xml/06444829.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06444829.pdf" title="Zbl 1392.94943 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.2015.02.007" aria-label="DOI for “A class of cyclic codes from two distinct finite fields”" title="10.1016/j.ffa.2015.02.007">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">Berndt, B.; Evans, R.; Williams, K., Gauss and Jacobi Sums (1997), John Wiley & Sons Company: John Wiley & Sons Company New York</td> </tr><tr> <td>[2]</td> <td class="space">Calderbank, A. 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