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Дата последнего поиска статьи во внешних источниках: 1 октября 2016 г.
Аннотация:In previous papers [4,5,6] we gave the
first 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 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.