Algorithms for majority decoding of group codes

We consider a problem of constructive description and justification of the algorithms necessary for a practical implementation of the majority decoder for group codes specified as left ideals of group algebras. In addition to the algorithms needed to implement a classical decoder of J. Massey, it is built a generalization of the classical decoder for codes with unequal protection of characters, which in some cases could be better than the classic one. For use as a classical decoder of J. Massey and its generalization to group codes it was developed an algorithm for constructing decoding trees that lie at the core of these algorithms for majority decoding. Because group codes are defined as left ideals of group algebras, the decoding algorithm for constructing decoding trees allows to build all decoding trees from one tree. On the basis of the generalized decoding algorithm it was developed an algorithm for decoding group codes induced on the subgroup. Application of the developed decoders was illustrated by an example of Reed-Muller-Berman codes and group codes induced by them on a non-Abelian group of affine transformations. In particular, for Reed–Muller–Berman code description and justification of the algorithm for constructing one decoding tree are provided. This three is used for constructing all decoding trees and then it is a built decoder for Reed–Muller–Berman codes and codes induced by them.