A Graph-Theoretical Method for Decoding Some Group MLD-Codes

Abstract: We construct the class of majority-logical decodable group codes using a method forcombining the codes that are based on the tensor product and the sum of codes. The constructionof this class rests on the Kasami–Lin technique, which allows us to consider not only individualcodes but also families of codes and utilizes the M-orthogonalityconstruction presented by Massey that is important for the majority-logical decodable codes. Thecodes under study are ideals in group algebras over, generally speaking, noncommutative finitegroups. Some algorithmic model of the majority decoding for the codes under consideration isdeveloped that is based on the graph-theoretic approach. An important part of this model is theconstruction of a special decoding graph for decoding one coordinate of a noisy codewordcorresponding to this graph. The group properties of the codes enable us to quickly find decodinggraphs for the remaining coordinates. We develop some decoding algorithm that corrects theerrors in all coordinates of the noisy codeword using this decoding graphs. As an example offamilies of group codes, we give the Reed–Muller binary codes important in cryptography. Thecode cryptosystems are considered as an alternative to the number-theoretic cryptosystems widelyused at present since they are resistant to attacks by quantum computers. The relevance of theproblem under consideration lies in the fact that the use of group codes and their variouscombinations is currently one of the promising ways to increase the stability of code cryptosystemsbecause enables us to construct new codes with a complex algebraic structure, which positivelyaffects the stability of the code cryptosystems. © 2020, Pleiades Publishing, Ltd.