Near-rings (MDS)- and Laguerre codes
type: Item
instance of: scholarly article
title: Near-rings (MDS)- and Laguerre codes
zbMATH Open document ID: 0708.94023
DOI: 10.1007/BF01230364
author: Helmut Karzel
author: Alan Oswald
published in: Journal of Geometry
publication date: 1990-01-01T00:00:00Z
review text: The authors study the construction of certain MDS-codes from near rings. Before we can give a more detailed description of their work, we have to caution the reader that their terminology and notation differs from what is usual in Coding Theory. The length of a code (usually denoted n) is here denoted by k (which usually denotes the dimension (in the linear case)). More annoyingly, the authors use the term MDS-code only for what one would usually call an MDS-code of dimension 2; MDS-codes of dimension 3 are what the authors choose to call Laguerre codes. The authors now discuss the construction of these two types of MDS-codes from near-rings, which includes the nonlinear case. The first part of the paper deals with the case of ``dimension'' 2 and contains nothing new: It is well-known that an MDS-code of length \(t+2\) over an alphabet of size q with \(q^ 2\) code words exists if and only if there are t mutually orthogonal Latin squares of order q [see e.g. \textit{J. Dénes} and \textit{A. D. Keedwell}, ``Latin squares and their applications'' (Budapest 1974; Zbl 0283.05014), p. 355]. As the authors themselves remark on p. 113, their near ring approach is equivalent to using a finite group N with a group G of fixed point free automorphisms. On the other hand, it is also well-known that such a group can be used to construct sets of MOLS by Mann's automorphism method [due to \textit{H. B. Mann}, ``The construction of orthogonal Latin squares'', Ann. Math. Stat. 13, 418-423 (1942; Zbl 0060.027) and ``On the construction of sets of mutually orthogonal Latin squares'', ibid. 14, 401-414 (1943; Zbl 0060.027)]. Moreover, the possible sizes of such sets (which determine the length of the resulting codes) are known for all abelian groups [see \textit{R. A. Bayley} and \textit{D. Jungnickel}, J. Comb. Theory, Ser. A 55, No.1, 1-13 (1990)]. In any case, the use of arbitrary sets of MOLS leads, of course, to better results, i.e. codes with larger length. In the second part of t…
Mathematics Subject Classification ID: 94B99
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 51E99
zbMATH DE Number: 4164743
zbMATH Keywords: MDS-codes from near rings
zbMATH Keywords: Laguerre codes
zbMATH Keywords: mutually orthogonal Latin squares
zbMATH Keywords: orthogonal arrays
reviewed by: Dieter Jungnickel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4164743