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The Structure of One Weight Linear and Cyclic Codes Over Z2^r x (Z2+uZ2)^s [article]

Ismail Aydogdu
<span title="2017-04-24">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently.  ...  Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages compared to Z2Z4-additive codes.  ...  In [9] , Carlet determined one weight linear codes over Z 4 and in [23] , Wood studied linear one weight codes over Z m .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1704.07212v1">arXiv:1704.07212v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eg6hss6pfnbfjgqsewu2gofnzi">fatcat:eg6hss6pfnbfjgqsewu2gofnzi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200929054431/https://arxiv.org/pdf/1704.07212v1.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/dc/0d/dc0d674f34e41f2481a232c5c72ab420d7b3b464.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1704.07212v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Two-weight and three-weight linear codes constructed from Weil sums

Tonghui Zhang, Hong Lu, Shudi Yang
<span title="">2022</span> <i title="American Institute of Mathematical Sciences (AIMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7mvec7wnufaoneapm2vcaq2wny" style="color: black;">Mathematical Foundations of Computing</a> </i> &nbsp;
In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets.  ...  <p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes.  ...  An [n, k, d] linear code C of length n over F q is a k-dimensional linear subspace of F n q with minimum (Hamming) distance d.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3934/mfc.2021041">doi:10.3934/mfc.2021041</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/b4sbytwm6zblndx7llmylaxvye">fatcat:b4sbytwm6zblndx7llmylaxvye</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220128001141/https://www.aimsciences.org/article/exportPdf?id=7efdd28e-ff33-40a5-b006-8628e900bfb0" 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/ef/f5/eff55652775bd119c1f4146480d3825a104a57fd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3934/mfc.2021041"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

The algebra of the finite Fourier transform and coding theory

R. Tolimieri
<span title="1985-01-01">1985</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes.  ...  The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut.  ...  A linear code in V or a linear code of length n over Z/p is any subspace C of V. The dimension of the linear code C in V is its dimension as a vector space over Z/p.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-1985-0766218-x">doi:10.1090/s0002-9947-1985-0766218-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qtxm4zzmirgdbg76ksshjoto3u">fatcat:qtxm4zzmirgdbg76ksshjoto3u</a> </span>
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The Algebra of the Finite Fourier Transform and Coding Theory

R. Tolimieri
<span title="">1985</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes.  ...  The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut.  ...  A linear code in V or a linear code of length n over Z/p is any subspace C of V. The dimension of the linear code C in V is its dimension as a vector space over Z/p.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2000409">doi:10.2307/2000409</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/72ji64gmv5dkjgqkdjqp45mxca">fatcat:72ji64gmv5dkjgqkdjqp45mxca</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170829075042/http://www.ams.org/journals/tran/1985-287-01/S0002-9947-1985-0766218-X/S0002-9947-1985-0766218-X.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/61/0f/610fe8baeaa4fe182b7fb9370b8d7118fd45b22b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2000409"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> jstor.org </button> </a>

Perfect reconstruction filter banks for HDTV representation and coding

Martin Vetterli, Jelena Kovačević, Didier J. Legall
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lbq64z5tgbddzjqwttcy2pxhsi" style="color: black;">Signal processing. Image communication</a> </i> &nbsp;
Subband decomposition of HDTV signals is important both for representation purposes (to create compatible subchannels) and for coding (several proposed compression schemes include some subband division  ...  Then we concentrate on the special case of quincunx subsampling and derive filter banks to go from progressive to interlaced scanning (with a highpass which contains deinterlacing information) as well  ...  One nice property is the easy control one has over the filter coefficients, which is useful in order to derive low complexity filters.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0923-5965(90)90011-6">doi:10.1016/0923-5965(90)90011-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lvyylaoodbfehmoytiajzng3im">fatcat:lvyylaoodbfehmoytiajzng3im</a> </span>
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A class of two or three weights linear codes and their complete weight enumerators [article]

Dabin Zheng, Qing Zhao, Xiaoqiang Wang, Yan Zhang
<span title="2020-09-10">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
forms and some special Weil sums over finite fields.  ...  In the past few years, linear codes with few weights and their weight analysis have been widely studied.  ...  linear codes over Galois rings.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.03879v2">arXiv:1911.03879v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3w4mwu26ozgqhcdst3t32tmdfy">fatcat:3w4mwu26ozgqhcdst3t32tmdfy</a> </span>
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Two-weight and three-weight linear codes based on Weil sums

Gaopeng Jian, Zhouchen Lin, Rongquan Feng
<span title="">2019</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vppj2ymhunarjnozole4nystdi" style="color: black;">Finite Fields and Their Applications</a> </i> &nbsp;
Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs.  ...  Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound.  ...  Hence, the weight distribution attracts many attentions in coding theory and plenty of work focuses on the determination of the weight distributions of linear codes [8, 11, 14, 20, 23, 26, 27, 22] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ffa.2019.02.001">doi:10.1016/j.ffa.2019.02.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t7v6lkrbgfd3van5t6oirjjzjm">fatcat:t7v6lkrbgfd3van5t6oirjjzjm</a> </span>
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Page 4665 of Mathematical Reviews Vol. , Issue 93h [page]

<span title="">1993</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 main purpose of the paper is to study the subsets P and D and linear codes related to an S(2,k,v). It is shown that (H,®,-,Z2) is a subspace of (P(S),4,-,Z2).  ...  codes are codes of projective geometries over GF(2).  ... 
<span class="external-identifiers"> </span>
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Two classes of linear codes from Weil sums

Hong Lu, Shudi Yang
<span title="">2020</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/q7qi7j4ckfac7ehf3mjbso4hne" style="color: black;">IEEE Access</a> </i> &nbsp;
If m d ≡ 0 (mod 4), by Lemma 6, if (6) is not solvable over F q Ω 2 = 0 (12) and if (6) is solvable over F q Ω 2 = −p s+d z2∈F * p χ(−z 2 a γ p u +1 a,b )ζ z2a p = −p s+d z2∈F * p ζ z2a Tr(γ p u  ...  An [n, k, d H ] linear code C over F p is a k-dimensional subspace of F n p with minimum Hamming distance d H . Let A i denote the number of codewords with Hamming weight i in a code C.  ...  Lemma 13 [9, Lemma 13] Every nonzero codeword of a linear code C over F p is minimal provided that where w min and w max denote the minimum and maximum nonzero weights in code C, respectively. A.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/access.2020.3028141">doi:10.1109/access.2020.3028141</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y7xdrwpwprg2hifbdsx2gb2uka">fatcat:y7xdrwpwprg2hifbdsx2gb2uka</a> </span>
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No-zero-entry, square, single-symbol ML Decodable STBCs

Ankita Pramanik, B. Sundar Rajan
<span title="">2010</span> <i title="IEEE"> 2010 National Conference On Communications (NCC) </i> &nbsp;
Space-time block codes based on orthogonal designs (ODs) are used for wireless communications with multiple transmit antennas which can achieve full transmit diversity along single-symbol maximum-likelihood  ...  Due to the zero entries in the design, the transmitting antenna need to be switched on and off imposing severe hardware constraints.  ...  The STBC obtained by letting each indeterminate to take all possible values from a complex constellation A is called a linear STBC over A.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ncc.2010.5430236">doi:10.1109/ncc.2010.5430236</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i2uloxxsonaibfmmkx5xuwpyzi">fatcat:i2uloxxsonaibfmmkx5xuwpyzi</a> </span>
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Construction of minimal linear codes with few weights from weakly regular plateaued functions

AHMET SINAK
<span title="2022-01-01">2022</span> <i title="The Scientific and Technological Research Council of Turkey (TUBITAK-ULAKBIM) - DIGITAL COMMONS JOURNALS"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3i63uqcqmvbdrbc6b26krcncii" style="color: black;">Turkish Journal of Mathematics</a> </i> &nbsp;
The construction of (minimal) linear codes from functions over finite fields has been greatly studied in the literature, since determining the parameters of codes based on functions is rather easy due  ...  In this paper, we derive 3 -weight and 4 -weight linear codes from weakly regular plateaued functions in the recent construction method of linear codes over the odd characteristic finite fields.  ...  Introduction There are many construction methods for linear codes, one of which is derived from functions over finite fields.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.55730/1300-0098.3135">doi:10.55730/1300-0098.3135</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3fnju6fyljavtboowb456wg24y">fatcat:3fnju6fyljavtboowb456wg24y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220525164140/https://journals.tubitak.gov.tr/cgi/viewcontent.cgi?article=3135&amp;context=math" 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/d2/b5/d2b5c7691f870ea2a12c10abff908aa9e984c663.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.55730/1300-0098.3135"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Page 5420 of Mathematical Reviews Vol. , Issue 2002G [page]

<span title="">2002</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;
Applying the analog of Construction A to a Type II code C of length k over Z2», a lattice A(C) in R* is obtained.  ...  Summary: “We determine all linear cyclic codes over Z4 of odd length whose Gray images are linear codes (or, equivalently, whose Nechaev-Gray images are linear cyclic codes, or which are them- selves linear  ... 
<span class="external-identifiers"> </span>
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Mode shape corrections for dynamic response to wind

J.D. Holmes
<span title="">1987</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2u4ircv5w5fbrisrqcnwdangaa" style="color: black;">Engineering structures</a> </i> &nbsp;
A correction factor, for nonlinear mode shapes, to the generalized force spectra in the first mode of vibration produced by wind action on tall structures, is proposed.  ...  Again, these methods are invariably based on data appropriate to a linear mode shape, and mode shape corrections are useful for these methods to broaden their applicability.  ...  height and R(z I -z 2 , n) is a cross-correlation function which depends on the separation distance, zl -z2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0141-0296(87)90017-4">doi:10.1016/0141-0296(87)90017-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g4yql6qax5dl5mdfevarikpamm">fatcat:g4yql6qax5dl5mdfevarikpamm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170811024554/http://www.jdhconsult.com/warreenpublishing/esmodeshapecorrections1987.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/83/99/83992b31e9400dec4fd20dcff5b8d25163f1ae97.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0141-0296(87)90017-4"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

A New Lower Bound for the Football Pool Problem for Six Matches

Patric R.J. Östergård, Alfred Wassermann
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
In the football pool problem one wants to minimize the cardinality of a ternary code, C F n 3 ; with covering radius one, and the size of a minimum code is denoted by s n : The smallest unsettled case  ...  The covering property of the code to be constructed can be expressed using a set of linear (in)equalities.  ...  INTRODUCTION The football pool problem is one of the oldest and most famous problems in combinatorial coding theory.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jcta.2002.3260">doi:10.1006/jcta.2002.3260</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/u2e2wik2qfgxhdzebczen2yqde">fatcat:u2e2wik2qfgxhdzebczen2yqde</a> </span>
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Page 4665 of Mathematical Reviews Vol. , Issue 97G [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;
+x +1)g(x) are generator polynomials of linear cyclic codes c(x) and c.yp(x) over GF(2) respectively; then Csub(x) is a subcode of c(x).  ...  This paper also determines the exact minimum Lee distance of the linear codes over Z, that are obtained from the extended binary two- and three-error correcting BCH codes by Hensel lifting.  ... 
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