Near-rings (MDS)- and Laguerre codes
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 their paper, the authors consider an analogous approach for the case of ``dimension'' 3. Here it is known that such MDS-codes correspond to orthogonal arrays of strength t, and the author's approach corresponds to finding an analogue of Mann's automorphism method for this case. While the conditions obtained here seem to be new, their interest would certainly have been enhanced by exhibiting at least one non-trivial example.
scientific article; zbMATH DE number 4164743