Let p = 3 be any prime. In this paper, we completely determine symbol-triple distance of all γ-constacyclic codes of length 3p s over the finite commutative chain ring R = F p m + uF p m , where γ is a unit of R which is not a cube in F p m . We give the necessary and sufficient condition for a symbol-triple γ-constacyclic code to be an MDS symbol-triple code. Using that, we establish all MDS symbol-triple γconstacyclic codes of length 3p s over R. Some examples of the symbol-triple distance of γ-constacyclic codes of length 3p s over R are provided. We also list some new MDS symbol-triple γ-constacyclic codes of length 3p s over R, where γ is not a cube in F p m . INDEX TERMS Constacyclic codes, dual codes, chain rings, MDS symbol-triple codes, symbol-triple codes.
Let p = 5 be any odd prime. Using the algebraic structures of all cyclic codes of length 5p s over the finite commutative chain ring R = F p m + uF p m , in this paper, the exact values of Hamming distances of all cyclic codes of length 5p s over R are established. As an application, we identify all maximal distance separable cyclic codes of length 5p s . INDEX TERMS Constacyclic codes, cyclic codes, dual codes, chain rings, Hamming distance, Singleton bound, MDS codes.
Let p≠3 be any prime. In this paper, we compute symbol-pair distance of all γ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm, where γ is a unit of R which is not a cube in Fpm. We give the necessary and sufficient condition for a symbol-pair γ-constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair γ-constacyclic codes of length 3ps over R. Some examples of the symbol-pair distance of γ-constacyclic codes of length 3ps over R are provided.
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