Linear complementary dual codes over rings

By using linear algebra over finite commutative rings, we will present some judging criterions for linear complementary dual (LCD) codes over rings, in particular, free LCD codes over finite commutative rings are described. By using free LCD codes over finite commutative rings and the Chinese Remainder Theorem, LCD codes over semi-simple rings are constructed and the equivalence of free codes and free LCD codes is given. In addition, all the possible LCD codes over chain rings are determined. We also generalize the judging criterion for cyclic LCD codes over finite fields to cyclic LCD codes over chain rings. Based on the above results and the Chinese Remainder Theorem, we also present results for LCD codes over principal ideal rings.

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