On symbol-pair distances of repeated-root constacyclic codes of length $$2p^s$$ over $${\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}$$ and MDS symbol-pair codes

Abstract

Let ℜ = p m + u p m with u 2 = 0 , where m, s are positive integers and p is an odd prime. For any invertible element of ℜ , the symbol-pair distances of all -constacyclic codes of length 2p s over ℜ are completely obtained. We identify all symbolpair Maximum Distance Separable (MDS) constacyclic codes of length 2p s over ℜ . As examples, many new symbol-pair codes, as well as symbol-pair MDS codes are constructed. Keywords Repeated-root codes • Symbol-pair MDS codes • Finite chain rings • Constacyclic codes • Symbol-pair distance Mathematics Subject Classification 94B15 • 11T71Authors' names are in alphabetical order. This is a part of Ph.D. work of Mr. M. K. Thakur.

Good codes from metacyclic groups

“…It is not difficult to see e 2 = e ande * = 2 • 2g 2 + 2 • g 4 + 2 • 2 • g 6 + 2 • g 8 + 2 = e.So, by Proposition 5.2, the code C generated by e is a LCD code of dimension of dimension 8 and weight 2 which are exactly the parameters of the best[10,8] code known.…”

“…When λ = 1, we have so called cyclic codes and, when λ = −1, we have negacyclic codes. Thus, constacyclic codes are generalization of cyclic and negacyclic codes and they have been studied for many authors ( [1], [2], [8]). Also, constacyclic codes can be realized as ideals in polinomial factor ring R[x]…”