A [6,3,4] code E(6) over an Abelian group A(4) with four elements is presented. E(6) is cyclic, unlike the [6,3,4] hexacode H-6 over GF(4). However, E(6) and H-6 are isomorphic when the latter is viewed as a group cod...
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A [6,3,4] code E(6) over an Abelian group A(4) with four elements is presented. E(6) is cyclic, unlike the [6,3,4] hexacode H-6 over GF(4). However, E(6) and H-6 are isomorphic when the latter is viewed as a group code. Differences and similarities between E(6) and H-6 are discussed. A dual code of E(6) is presented. Some binary codes, among them the [24,12,8] Golay, are derived with the aid of E(6). A related cyclic [4,2,3] code E(4)* is applied to construct the Nordstrom-Robinson code. E(6) is the smallest member of a class of [2k,k,4] cyclic and reversible codes over A(4). Another class oi cyclic and reversible codes of length 2l + 1: l greater than or equal to 2 and minimum distance 3 over A(4) is also presented.
In this note we answer some questions on LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgr...
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In this note we answer some questions on LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ LCD }$$\end{document} group codes posed in de la Cruz and Willems (Des codes Cryptogr 86:2065-2073, 2018) and (Vietnam J Math 51:721-729, 2023). Furthermore, over prime fields we determine completely the p-part of the divisor of an LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ LCD }$$\end{document} group code. In addition we present a natural construction of nearly LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ LCD }$$\end{document} codes.
We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension 4, we show that there is no quasi-...
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We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension 4, we show that there is no quasi-group code whose permutation automorphism group is isomorphic to C-2. By generalizing the method we use to prove this result, we obtain results on the structure of putative extremal self-dual [16, 36, 72] and [20, 48, 96] codes in the presence of an involutory permutation automorphism.
Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dim(F)(C) 2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite ...
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Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dim(F)(C) <= 3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when char F > 2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4. Published by Elsevier Inc.
group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fi...
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group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fields of any characteristic are asymptotically good. (C) 2020 Elsevier Inc. All rights reserved.
We investigate and characterize ideals in a group algebra KG which have complementary duals, i.e., ideals C in KG which satisfy In the special case that G is a cyclic group we get an early result of Yang and Massey as...
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We investigate and characterize ideals in a group algebra KG which have complementary duals, i.e., ideals C in KG which satisfy In the special case that G is a cyclic group we get an early result of Yang and Massey as an easy consequence.
We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n, n - 3, 3](q)- codes over F-q for n = 2q and n = 3q. These codes are linearly optimal, i.e. have maximal di...
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We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n, n - 3, 3](q)- codes over F-q for n = 2q and n = 3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed-Solomon codes as ideals of the group ring F(q)G where G is an elementary abelian group of order q. (C) 2010 Elsevier Inc. All rights reserved.
Designing low coherence matrices and low-correlation frames is a point of interest in many fields including compressed sensing, MIMO communications and quantum measurements. The challenge is that one must control the ...
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ISBN:
(纸本)9781479904464
Designing low coherence matrices and low-correlation frames is a point of interest in many fields including compressed sensing, MIMO communications and quantum measurements. The challenge is that one must control the ((n)(2)) pairwise inner products between the frame elements. In this paper, we exploit the group code approach of David Slepian [1], which constructs frames using unitary group representations and which in general reduces the number of distinct inner products to n - 1. We demonstrate how to efficiently find optimal representations of cyclic groups, and we show how basic abelian groups can be used to construct tight frames that have the same dimensions and inner products as those arising from certain more complex nonabelian groups. We support our work with theoretical bounds and simulations.
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