Abstract

We construct the class of majority-logical decodable group codes using a method for combining the codes that are based on the tensor product and the sum of codes. The construction of this class rests on the Kasami–Lin technique, which allows us to consider not only individual codes but also families of codes and utilizes the \(M\)-orthogonality construction presented by Massey that is important for the majority-logical decodable codes. The codes under study are ideals in group algebras over, generally speaking, noncommutative finite groups. Some algorithmic model of the majority decoding for the codes under consideration is developed that is based on the graph-theoretic approach. An important part of this model is the construction of a special decoding graph for decoding one coordinate of a noisy codeword corresponding to this graph. The group properties of the codes enable us to quickly find decoding graphs for the remaining coordinates. We develop some decoding algorithm that corrects the errors in all coordinates of the noisy codeword using this decoding graphs. As an example of families of group codes, we give the Reed–Muller binary codes important in cryptography. The code cryptosystems are considered as an alternative to the number-theoretic cryptosystems widely used at present since they are resistant to attacks by quantum computers. The relevance of the problem under consideration lies in the fact that the use of group codes and their various combinations is currently one of the promising ways to increase the stability of code cryptosystems because enables us to construct new codes with a complex algebraic structure, which positively affects the stability of the code cryptosystems.

中文翻译：

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图论方法解码某些组MLD码

摘要

我们使用一种基于张量积和代码总和的组合代码的方法来构造多数逻辑可解码组代码的类。此类的构造基于Kasami-Lin技术，该技术使我们不仅可以考虑单个代码，还可以考虑代码族，并利用\（M \）-由Massey提出的正交结构，对于多数逻辑可解码代码很重要。在非代数有限群上，研究代号是群代数中的理想选择。基于图论方法，开发了一种针对所考虑代码的多数解码算法模型。该模型的重要部分是特殊解码图的构造，用于解码与该图相对应的噪声码字的一个坐标。代码的组属性使我们能够快速找到其余坐标的解码图。我们开发了一些解码算法，可以使用该解码图来纠正噪声码字的所有坐标中的错误。作为一组组码的示例，我们给出了Reed-Muller二进制码在密码学中很重要。代码密码系统被认为可以替代目前广泛使用的数论密码系统，因为它们可以抵抗量子计算机的攻击。正在考虑的问题的相关性在于，使用组代码及其各种组合目前是提高代码密码系统稳定性的有前途的方法之一，因为它使我们能够构建具有复杂代数结构的新代码，这肯定会影响代码密码系统的稳定性。