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New ring-linear codes from dualization in projective Hjelmslev geometries

DESIGNS codeS AND CRYPTOGRAPHY 2013年第1-3期66卷 39-55页

作者： Kiermaier, Michael Zwanzger, Johannes Univ Bayreuth Math Inst D-95440 Bayreuth Germany Siemens AG CT T DE IT1 D-81739 Munich Germany

In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. Applying it to the -preimages of the Kerdock codes and a related family of codes we will call Teichmuller codes, we get two new infinite series of codes and compute their symmetrized weight enumerators. In some cases, residuals of the original code give further interesting codes. The generalized Gray map translates our codes into ordinary, generally non-linear codes in the Hamming space. The obtained parameters include (58, 2(7), 28)(2), (60, 2(8), 28)(2), (114, 2(8), 56)(2), (372, 2(10), 184)(2) and (1988, 2(12), 992)(2) which provably have higher minimum distance than any linear code of equal length and cardinality over an alphabet of the same size (better-than-linear, BTL), as well as (180, 2(9), 88)(2), (244, 2(9), 120)(2), (484, 2(10), 240)(2) and (504, 4(6), 376)(4) where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).

关键词： ring-linear code Kerdock code Lee weight Homogeneous weight Galois ring Gray map Hjelmslev geometry

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Two-weight codes, graphs and orthogonal arrays

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DESIGNS codeS AND CRYPTOGRAPHY 2016年第2期79卷 201-217页

作者： Byrne, Eimear Sneyd, Alison Univ Coll Dublin Sch Math Sci Dublin 2 Ireland

We investigate properties of two-weight codes over finite Frobenius rings, giving constructions for the modular case. A -modular code (in: Honold T, Honold in Proceedings of the fifth international workshop on optimal codes and related topics, White Lagoon, Bulgaria, 2007) is characterized as having a generator matrix where each column appears with multiplicity for some . Generalizing (Delsarte in Discret Math 3:47-64, 1972) and (Byrne et al in Finite Fields Appl 18(4):711-727, 2012), we show that the additive group of a two-weight code satisfying certain constraint equations (and in particular a modular code) has a strongly regular Cayley graph and derive existence conditions on its parameters. We provide a construction for an infinite family of modular two-weight codes arising from unions of submodules with pairwise trivial intersection. The corresponding strongly regular graphs are isomorphic to graphs from orthogonal arrays.

关键词： ring-linear code Finite Frobenius ring Orthogonal array Strongly regular graph Homogeneous weight Two-weight code Modular code

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The weight distribution of codes over finite chain rings

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linear ALGEBRA AND ITS APPLICATIONS 2023年第1期675卷 90-105页

作者： Cavicchioni, Giulia Meneghetti, Alessio Univ Trento Dept Math Trento Italy

In this work, we determine new linear equations for the weight distribution of linear codes over finite chain rings. The identities are determined by counting the number of some special submatrices of the parity-check matrix of the code. Thanks to these relations we are able to compute the full weight distribution of codes with small Singleton defects, such as MDS, MDR and AMDR codes.& COPY;2023 Elsevier Inc. All rights reserved.

关键词： ring-linear code Weight distribution

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Properties of codes with two homogeneous weights

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FINITE FIELDS AND THEIR APPLICATIONS 2012年第4期18卷 711-727页

作者： Byrne, Eimear Kiermaier, Michael Sneyd, Alison Univ Coll Dublin Sch Math Sci Dublin Ireland Univ Coll Dublin Claude Shannon Inst Dublin Ireland Univ Bayreuth Math Inst Bayreuth Germany

Delsarte showed that for any projective linear code over a finite field GF(p(r)) with two nonzero Hamming weights w(1) < w(2) there exist positive integers u and s such that w(1) = p(s)u and w(2) = p(s)(u + 1). Moreover, he showed that the additive group of such a code has a strongly regular Cayley graph. Here we show that for any regular projective linear code C over a finite Frobenius ring with two integral nonzero homogeneous weights w(1) < w(2) there is a positive integer d, a divisor of vertical bar C vertical bar, and positive integer u such that w(1) = du and w(2) = d(u + 1). This gives a new proof of the known result that any such code yields a strongly regular graph. We apply these results to existence questions on two-weight codes. (C) 2012 Elsevier Inc. All rights reserved.

关键词： ring-linear code Homogeneous weight Weight distribution Two-weight code Character module Strongly regular graph Cayley graph

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On coloured constant composition designs

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DISCRETE MATHEMATICS 2009年第8期309卷 2410-2416页

作者： Greferath, Marcus Therkelsen, Ryan K. N Carolina State Univ Dept Math Raleigh NC 27695 USA Univ Coll Dublin Sch Math Sci Dublin 4 Ireland

This paper deals with a particular class of coloured designs: the incidence vectors of all blocks of these designs have the same composition, and the same is true for the incidence vectors of all points. For this reason, we call these designs constant composition designs, or CC-designs for short. We will derive necessary and sufficient conditions on the existence and conclude the presentation with a collection of examples. (c) 2008 Elsevier B.V. All rights reserved.

关键词： Coloured design ring-linear code Weight function Finite ring geometry

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codes from translation schemes on Galois rings of characteristic 4

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Electronic Notes in Discrete Mathematics 2013年 40卷 175-180页

作者： Kiermaier, Michael Institut für Mathematik Universität Bayreuth Bayreuth Germany

In [Thomas Honold. Two-intersection sets in projective Hjelmslev spaces. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pages 1807-1813, 2010], it has been shown that the Teichmüller point set in the projective Hjelmslev geometry PHG(R^k) over a Galois ring R of characteristic 4 with k odd is a two-intersection set. From this result, the parameters of the generated codes can be derived, see [Michael Kiermaier and Johannes Zwanzger. New ring-linear codes from dualization in projective Hjelmslev geometries. To appear in Des. codes Cryptogr. doi: 10.1007/s10623-012-9650-1, Fact 5.2]. The resulting Teichmüller codes have a high minimum distance. The key step in the proof of the two-weight property in [Thomas Honold. Two-intersection sets in projective Hjelmslev spaces. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pages 1807-1813, 2010.] is to show that for a certain supergroup σ of the Teichmüller units T in a Galois ring S of characteristic 4, the partition Aσ={ring-linear code,2S*,σ,S*\σ} induces a translation scheme on (S, +). We generalize these results by characterizing all supergroups σ of T such that Aσ induces a symmetric translation scheme. In turn, we get new two-intersection sets in projective Hjelmslev geometries and two new series Tq,k,s and Uq,k,s of R-linear codes. The series Tq,k,s generalizes the Teichmüller codes (special case s=0). The codes Uq,k,s are homogeneous two-weight codes. Application of the dualization construction to Tq,k,s yields another series Tq,k,s*. The Gray images of the codes Tq,k,s and Tq,k,s* have a higher minimum distance than all known Fq-linear codes of the same length and size. © 2013 Elsevier B.V.

关键词： Association scheme Homogeneous weight Projective Hjelmslev geometry ring-linear code Teichmüller group

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Bounds in the Lee metric and optimal codes

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FINITE FIELDS AND THEIR APPLICATIONS 2023年第1期87卷

作者： Byrne, Eimear Weger, Violetta Univ Coll Dublin Dublin 4 Dublin D04V1W8 Ireland Tech Univ Munich Theresienstr 90 D-80799 Munich Germany

In this paper we investigate known Singleton-like bounds in the Lee metric and characterize their extremal codes, which turn out to be very few. We then focus on Plotkin-like bounds in the Lee metric and present a new bound that extends and refines a previously known, and out-performs it in the case of non-free codes. We then compute the density of extremal codes with regard to the new bound. Finally we fill a gap in the characterization of Lee-equidistant codes. (c) 2022 Elsevier Inc. All rights reserved.

关键词： ring-linear code Lee distance Maximum Lee distance Bounds Constant weight codes

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