New Examples of Non-Abelian Group Codes

Abstract

AbstractIt has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring \(F_{5}S_{4}\). The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F 3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F 2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F 2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.KeywordsGroup codeAbelian group codeSemisimplicity