Document Zbl 1005.94023

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Also, the notions of self-dual codes over $\mathbb{Z}_4$ are given, simplifying the definition and properties of shadow codes. The authors define some even weight subcodes and give the properties for these groups of codes. The information known to us on the weight enumerators and the highest minimum weights of codes of length up to 24 are also collected.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A771">Demetre Voukalis (Athens)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1763655">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A94B05" title="MSC2020">94B05</a> </td> <td class="space"> Linear codes (general theory) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94B65" title="MSC2020">94B65</a> </td> <td class="space"> Bounds on codes </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aself-dual+codes">self-dual codes</a>; <a href="/?q=ut%3Ashadow+codes">shadow codes</a>; <a href="/?q=ut%3Aweight+enumerators">weight enumerators</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 1005.94023" data-ciurl="/ci/01763655" data-biburl="/bibtex/01763655.bib" data-amsurl="/amsrefs/01763655.bib" data-xmlurl="/xml/01763655.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01763655.pdf" title="Zbl 1005.94023 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/10.1006/ffta.2000.0312" aria-label="DOI for “Shadow codes over ${\mathbb Z}_4$”" title="10.1006/ffta.2000.0312">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="mathoverflow"> <h3>Online Encyclopedia of Integer Sequences:</h3> <a href="https://oeis.org/A66012">Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105681.</a><br> <a href="https://oeis.org/A66013">Number of codes having highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105688.</a><br> <a href="https://oeis.org/A66014">Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105682.</a><br> <a href="https://oeis.org/A66015">Number of codes having highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105682.</a><br> <a href="https://oeis.org/A66016">Highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.</a><br> <a href="https://oeis.org/A66017">Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.</a><br> <a href="https://oeis.org/A105681">Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.</a><br> <a href="https://oeis.org/A105682">Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.</a><br> <a href="https://oeis.org/A105688">Number of codes having highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.</a><br> <a href="https://oeis.org/A105689">Number of codes having highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.</a><br> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bannai, E.; Dougherty, S. T.; Harada, M.; Oura, M., Type II codes, even unimodular lattices and invariant rings, IEEE Trans. Inform. Theory, 45, 1194-1205 (1999) · <a href="/0958.94042" class="nowrap">Zbl 0958.94042</a></td> </tr><tr> <td>[2]</td> <td class="space">Bonnecaze, A.; Solé, P.; Calderbank, A. R., Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, 41, 366-377 (1995) · <a href="/0822.94009" class="nowrap">Zbl 0822.94009</a></td> </tr><tr> <td>[3]</td> <td class="space">Bonnecaze, A.; Solé, P.; Bachoc, C.; Mourrain, B., Type II codes over $Z_4$, IEEE Trans. Inform. Theory, 43, 969-976 (1997) · <a href="/0898.94009" class="nowrap">Zbl 0898.94009</a></td> </tr><tr> <td>[4]</td> <td class="space">Brualdi, R. A.; Pless, V., Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 37, 1222-1225 (1991) · <a href="/0733.94018" class="nowrap">Zbl 0733.94018</a></td> </tr><tr> <td>[5]</td> <td class="space">Conway, J. H.; Sloane, N. J.A., On the enumeration of lattices of determinant one, J. Number Theory, 15, 83-94 (1982) · <a href="/0496.10023" class="nowrap">Zbl 0496.10023</a></td> </tr><tr> <td>[6]</td> <td class="space">Conway, J. H.; Sloane, N. J.A., A new upper bound on the minimum distance of self-dual codes, IEEE Trans. Inform. Theory, 36, 1319-1333 (1990) · <a href="/0713.94016" class="nowrap">Zbl 0713.94016</a></td> </tr><tr> <td>[7]</td> <td class="space">Conway, J. H.; Sloane, N. J.A., Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A, 62, 30-45 (1993) · <a href="/0763.94018" class="nowrap">Zbl 0763.94018</a></td> </tr><tr> <td>[8]</td> <td class="space">Dougherty, S. T., Shadow codes and weight enumerators, IEEE Trans. Inform. Theory, 41, 762-768 (1995) · <a href="/0824.94020" class="nowrap">Zbl 0824.94020</a></td> </tr><tr> <td>[9]</td> <td class="space">Fields, J.; Gaborit, P.; Leon, J.; Pless, V., All self-dual $Z_4$ codes of length 15 or less are known, IEEE Trans. Inform. Theory, 44, 311-322 (1998) · <a href="/0899.94027" class="nowrap">Zbl 0899.94027</a></td> </tr><tr> <td>[10]</td> <td class="space">Gulliver, T. A.; Harada, M., Certain self-dual codes over $Z_4$ and the odd Leech lattice, Lecture Notes in Computer Sciences, 1225 (1997), Springer-Verlag: Springer-Verlag New York, p. 130-137 · <a href="/1042.94534" class="nowrap">Zbl 1042.94534</a></td> </tr><tr> <td>[11]</td> <td class="space">Hammons, A. R.; Kumar, P. V.; Calderbank, A. R.; Sloane, N. J.A.; Solé, P., A linear construction for certain Kerdock and Preparata codes, Bull. Amer. Math. Soc., 29, 218-222 (1993) · <a href="/0783.94026" class="nowrap">Zbl 0783.94026</a></td> </tr><tr> <td>[12]</td> <td class="space">Harada, M., Self-Dual Codes, Designs and Unimodular Lattices (1997), Okayama University · <a href="/0933.05028" class="nowrap">Zbl 0933.05028</a></td> </tr><tr> <td>[13]</td> <td class="space">Harada, M.; Gaborit, P.; Solé, P., Self-dual codes over $Z_4$ and unimodular lattices: A survey, Algebras and Combinatorics (Hong Kong, 1997), 255-275 (1999) · <a href="/0998.11035" class="nowrap">Zbl 0998.11035</a></td> </tr><tr> <td>[14]</td> <td class="space">Klemm, M., Über die Identät von MacWilliams für die Gewichtsfunktion von Codes, Arch. Math., 49, 400-406 (1987) · <a href="/0606.94005" class="nowrap">Zbl 0606.94005</a></td> </tr><tr> <td>[15]</td> <td class="space">Mallows, C.; Sloane, N. J.A., An upper bound for self-dual codes, Inform. Control, 22, 188-200 (1973) · <a href="/0254.94011" class="nowrap">Zbl 0254.94011</a></td> </tr><tr> <td>[16]</td> <td class="space">Pless, V.; Solé, P.; Qian, Z., Cyclic self-dual $Z_4$-codes, Finite Fields Appl., 3, 48-69 (1997) · <a href="/1053.94573" class="nowrap">Zbl 1053.94573</a></td> </tr><tr> <td>[17]</td> <td class="space">Rains, E. M., Optimal self-dual codes over $Z_4$, Discrete Math., 203, 215-228 (1999) · <a href="/1097.94525" class="nowrap">Zbl 1097.94525</a></td> </tr><tr> <td>[18]</td> <td class="space">Rains, E. M.; Sloane, N. J.A., Self-dual codes, (Pless, V.; Brualdi, R.; Huffman, C., Handbook of Coding Theory (1998), Elsevier: Elsevier Amsterdam/New York) · <a href="/0936.94017" class="nowrap">Zbl 0936.94017</a></td> </tr><tr> <td>[19]</td> <td class="space">Tsai, H.-T., Existence of some extremal self-dual codes, IEEE Trans. Inform. Theory, 38, 1829-1833 (1992) · <a href="/0759.94008" class="nowrap">Zbl 0759.94008</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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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