New examples of non-abelian group codes
Cristina García Pillado, Santos González, Victor Markov, Consuelo Martı́nez, Alexandr A. Nechaev · Advances in Mathematics of Communications · 2016
In previous papers [4,5,6] we gave thefirst example of a non-abelian group code using the group ring$F_5S_4$. It is natural to ask if it is really relevant that thegroup ring is semisimple. What happens if the field hascharacteristic $2$ or $3$? We have addressed this question, withcomputer help, proving that there are also examples of non-abeliangroup codes in the non-semisimple case. The results show someinteresting differences between the cases of characteristic $2$and $3$. Furthermore, using the group $SL(2,F_3)$, we construct anon-abelian group code over $F_2$ of length $24$, dimension $6$and minimal weight $10$. This code is optimal in the followingsense: every linear code over $F_2$ with length $24$ and dimension$6$ has minimum distance less than or equal to $10$. In the caseof abelian group codes over $F_2$ the above value for theminimum distance cannot be achieved, since the minimum distanceof a binary abelian group code with the given length and dimension6 is at most 8.