The Gray Image of Codes over Finite Chain Rings

Details The Gray Image of Codes over Finite Chain Rings The Gray Image of Codes over Finite Chain Rings

2009-07-20

arxiv.org/abs/0907.3397v4

Mathematics

Rings and Algebras

Submit to Discrete Mathematics

Scientific paper

The results of J. F. Qiann et al. [4] on $(1-\gamma)$-cyclic codes over finite chain rings of nilpotency index 2 are extended to $(1-\gamma^e)$-cyclic codes over finite chain rings of arbitrary nilpotency index $e+1$. The Gray map is introduced for this type of rings. We prove that the Gray image of a linear $(1 - \gamma^{e})$-cyclic code over a finite chain ring is a distance-invariant quasi-cyclic code over its residue field. When the length of codes and the characteristic of a ring are relatively prime, the Gray images of a linear cyclic code and a linear $(1+\gamma^e)$-cyclic code are permutatively to quasi-cyclic codes over its residue field.

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