Construction of self-orthogonal codes over a commutative non-unitary ring of order 25

초록

Codes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring (in the classification of Fine) of order where p is a prime are being considered. For (resp. ), three categories of codes over have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over . Using some related mass formulas and building-up constructions, classifications of these codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime and consider the ring . We introduce the notion of linear codes over . We also define the same three categories of linear -codes, study the structures of these -codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most 4 up to the monomial equivalence for a given type . Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over , mistakes in the classification of quasi self-dual codes over had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over for length and type and for length and type . We correct and improve such results.

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