New examples of non-abelian group codes.

In previous papers [4, 5, 6] we gave the first example of a nonabelian group code using the group ring F5S4. It is natural to ask if it is really relevant that the group ring is semisimple. What happens if the field has characteristic 2 or 3? We have addressed this question, 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. Furthermore, using the group SL(2, F-3), we construct a non-abelian group code over F-2 of length 24, dimension 6 and minimal weight 10. This code is optimal in the following sense: every linear code over F-2 with length 24 and dimension 6 has minimum distance less than or equal to 10. In the case of abelian group codes over F-2 the above value for the minimum distance cannot be achieved, since the minimum distance of a binary abelian group code with the given length and dimension 6 is at most 8.