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The nonexistence of some quaternary linear codes of dimension 5

Tatsuya Maruta
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The values of n q (k; d) are determined for all d only for some small values of q and k. For quaternary linear codes, n 4 (k; d) is known for k64 for all d [3,6,11].  ...  We prove the nonexistence of linear codes with parameters [400; 5; 299] 4 , [401; 5; 300] 4 , [405; 5; 303] 4 , [406; 5; 304] 4 , [485; 5; 363] 4 and [486; 5; 364] 4 attaining the Griesmer bound.  ...  To prove Theorem 1.1 (in Sections 6 -8) we characterize some quaternary linear codes of dimension four (in .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00413-1">doi:10.1016/s0012-365x(00)00413-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tccqdxj62rekhjyltgabdcufxe">fatcat:tccqdxj62rekhjyltgabdcufxe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190415004755/https://core.ac.uk/download/pdf/82782291.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b8/a0/b8a01737dd1c4cbfdbbd9ff09e16e5f1ca191789.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00413-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 8142 of Mathematical Reviews Vol. , Issue 99k [page]

<span title="">1999</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The possible weight hierarchies of such binary codes of dimension 4 are determined.” 99k:94046 94B05 Hamada, Noboru (J-OWU-AM; Osaka) On the nonexistence of some quaternary linear codes meeting the Griesmer  ...  The au- thor proves the nonexistence of these quaternary linear codes using the nonexistence of {48,11;4,4}, {52,12;4,4} and {64, 15; 4, 4}- minihypers, respectively.  ... 
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Optimal additive quaternary codes of low dimension [article]

Juergen Bierbrauer, Stefano Marcugini, Fernanda Pambianco
<span title="2020-07-10">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We determine the optimal parameters of additive quaternary codes of dimension k≤ 3. The most challenging case is dimension k=2.5.  ...  An additive quaternary [n,k,d]-code (length n, quaternary dimension k, minimum distance d) is a 2k-dimensional F_2-vector space of n-tuples with entries in Z_2× Z_2 (the 2-dimensional vector space over  ...  The optimal parameters of linear quaternary 3-dimensional codes are of course known: Proposition 2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.05482v1">arXiv:2007.05482v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wso7phaaivdnfp4ufbengspihq">fatcat:wso7phaaivdnfp4ufbengspihq</a> </span>
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Short Additive Quaternary Codes

JÜrgen Bierbrauer, Yves Edel, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
<span title="">2009</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
Among the results obtained in this work are the non-existence of [12, 7, 5]-codes and [12, 4.5, 7]-codes as well as the existence of a [13, 7.5, 5]−code.  ...  Index Terms-Linear codes, quaternary additive codes, binary projective spaces.  ...  An additive quaternary [n, k]-code C (length n, dimension k) is a 2k-dimensional subspace of F 2n 2 , where the coordinates come in pairs of two.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2008.2011447">doi:10.1109/tit.2008.2011447</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r6tjmwratnhsjnubvoy4jzhvba">fatcat:r6tjmwratnhsjnubvoy4jzhvba</a> </span>
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Editorial: 3rd International Castle Meeting on Coding Theory and Applications

Joaquim Borges, Mercè Villanueva, Victor Zinoviev
<span title="2012-12-08">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
Feulner prove the nonexistence of a [21, 14, 6]-code over F 4 and a [16, 5, 10]-code over F 5 , which implies also some new upper bounds for minimum distance of linear codes of given length and dimension  ...  Pernas, Pujol, and Villanueva study the order and structure of the permutation automorphism group of quaternary linear Hadamard codes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10623-012-9775-2">doi:10.1007/s10623-012-9775-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pa276hqqgja4fjsq44biyataja">fatcat:pa276hqqgja4fjsq44biyataja</a> </span>
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The nonexistence of an additive quaternary [15,5,9]-code [article]

Daniele Bartoli, Juergen Bierbrauer, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
<span title="2013-08-09">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that no additive [15,5,9]_4-code exists. As a consequence the largest dimension k such that an additive quaternary [15,k,9]_4-code exists is k=4.5.  ...  An additive quaternary [n, k] 4 -code C (length n, dimension k) is a 2k-dimensional subspace of F 2n 2 , where the coordinates come in pairs of two.  ...  As a linear quaternary [15, 4, 10]code exists (it is derivable from the [17,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1308.2108v1">arXiv:1308.2108v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3q5evic3lra7tnk5xs6vp5p6ni">fatcat:3q5evic3lra7tnk5xs6vp5p6ni</a> </span>
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Page 7043 of Mathematical Reviews Vol. , Issue 91M [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k < n(n—yw+1).  ...  7043 dimensional linear space of n x n matrices over F such that every nonzero matrix in C has rank > yu.  ... 
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Quaternary Hermitian linear complementary dual codes [article]

Makoto Araya, Masaaki Harada, Ken Saito
<span title="2019-12-28">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we give some conditions for the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights.  ...  As an application, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes.  ...  The authors would like to thank the anonymous referees for the useful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.07517v2">arXiv:1904.07517v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fron32eqsbafbkbsexdyrmvxzi">fatcat:fron32eqsbafbkbsexdyrmvxzi</a> </span>
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Optimal quaternary linear codes of dimension five

I. Boukliev, R. Daskalov, S. Kapralov
<span title="">1996</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
A survey of the results of recent work on bounds for quaternary linear codes in dimensions four and five is made and a table with lower and upper bounds for drl ( n , 5) is presented. distance of a q-ary  ...  code r-nonexistence of an [ n~ k ; d ; 41-code via its residual code d-nonexistence of an [ n , k , d ; 41-code follows from the nonexistence of its dual code For all the others lower bounds ( 1 5 n 5  ...  A linear code C of length n and dimension k over GF ( q ) is a k-dimensional subspace of V ( n , 4). Such a code is called [ T I . k . d: q] -code if its minimum Hamming distance is d .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/18.508846">doi:10.1109/18.508846</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/je2ktxveobhcdneg4tglxmm42i">fatcat:je2ktxveobhcdneg4tglxmm42i</a> </span>
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On the minimum length of quaternary linear codes of dimension five

Ivan N. Landjev, Tatsuya Maruta
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In this paper we prove the nonexistence of quaternary linear codes with parameters [190, 5, 141], [239, 5, 178], [275, 5, 205], [288, 5, 215], [291, 5, 217] and [488, 5, 365].  ...  Let n,(k, d) be the smallest integer n for which there exists a linear code of length n, dimension k and minimum distance d, over the q-element field.  ...  of some quaternary linear codes with a well- known geometric method.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(98)00354-9">doi:10.1016/s0012-365x(98)00354-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cw6g35whsjfyzmog22o2clxtby">fatcat:cw6g35whsjfyzmog22o2clxtby</a> </span>
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Small additive quaternary codes

A Blokhuis, A.E Brouwer
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
We determine the parameters of the optimal additive quaternary codes of length at most 12 over Z 2 × Z 2 .  ...  Or again, how many lines one can pick in a binary projective space such that no hyperplane contains more than m of them.  ...  Some of the codes found improve on the best quaternary codes known. An additive code over Q is a vector space over F 2 , of some dimension, k say, and hence has size 2 k .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0195-6698(03)00096-9">doi:10.1016/s0195-6698(03)00096-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3bzb2wvujbhrviirhz6v6g553i">fatcat:3bzb2wvujbhrviirhz6v6g553i</a> </span>
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Orthogonal arrays of strength 3 and small run sizes

Andries E. Brouwer, Arjeh M. Cohen, Man V.M. Nguyen
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rllg36he2jdbba4ztp4efgsxqq" style="color: black;">Journal of Statistical Planning and Inference</a> </i> &nbsp;
All mixed (or asymmetric) orthogonal arrays of strength 3 with run size at most 64 are determined.  ...  A [n, k, d] q code is a linear code of word length n, dimension k, and minimum distance d. The code words of the dual code (that has dimension n − k) form an OA(N, q n , d − 1) with N = q n−k .  ...  First proof: an OA(27, 3 5 , 3) would be a ternary code (not necessarily linear) of size 27, word length 5, and dual distance at least 4, contradicting the Delsarte linear programming bound.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jspi.2004.12.012">doi:10.1016/j.jspi.2004.12.012</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yfan3qrfivcerjo7dt5yddy7xi">fatcat:yfan3qrfivcerjo7dt5yddy7xi</a> </span>
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Page 1380 of Mathematical Reviews Vol. , Issue 97B [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Hall (1-MIS; East Lansing, MI) 97b:94032 94B05 05B30 62K15 94B65 Hamada, Noboru (J-OWU-AM; Osaka) The nonexistence of some quaternary linear codes meeting the Griesmer bound and the bounds for n3(5,d),  ...  Four non-existence results for ternary linear codes; Chapter 8. Some new results for ternary linear codes of dimension 5 and 6; Chapter 9.  ... 
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Adding a parity-check bit

J. Simonis
<span title="">2000</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
and minimum distance to be extendable to a code of the same dimension, length + 1, and minimum distance Index Terms-Code extension, linear codes, parity-check bit.  ...  [15] , "An improvement of the Griesmer bound for some classes of distances," Probl. Inform I.  ...  INTRODUCTION In [12], the first author introduced a very general description of cocyclic codes in order to demonstrate the previously unrecognized (and well-hidden) presence of cocycles in several code  ... 
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Optimal Quaternary Hermitian LCD codes [article]

Liangdong Lu and Xiuzhen Zhan and Sen Yang and Hao Cao
<span title="2020-10-20">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Results including tables 3 of the best-known quaternary Hermitian LCD codes of any length n ≤ 25 with corresponding dimension k are presented.  ...  In addition, Many of these quaternary Hermitian LCD codes given in this paper are optimal which are saturating the lower or upper bound of Grassl's codetable in and some of them are nearly optimal.  ...  Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No.11801564.  ... 
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