MDS Symbol-Pair Repeated-Root Constacylic Codes of Prime Power Lengths Over $\mathbb F_{p^m}+ u \mathbb F_{p^m}$

Abstract

MDS codes have the highest possible error-detecting and error-correcting capability among codes of given length and size. Let p be any prime, and s, m be positive integers. Here, we consider all constacyclic codes of length p s over the ring R = F p m + uF p m (u 2 = 0). The units of the ring R are of the form α + uβ and γ, where α, β, γ ∈ F* p m, which provides p m (p m - 1) constacyclic codes. We acquire that the (α + uβ)-constacyclic codes of ps length over R are the ideals 〈(α 0 x - 1) j 〉, 0 ≤ j ≤ 2 p s , of the finite chain ring R[x]/〈x ps - (α + uβ)〉 and the γ-constacyclic codes of ps length over R are the ideals of the ring R[x]/〈x ps - γ〉 which is a local ring with the maximal ideal 〈u, x - γ 0 〉, but it is not a chain ring. In this paper, we obtain all MDS symbol-pair constacyclic codes of length ps over R. We deduce that the MDS symbol-pair constacyclic codes are the trivial ideal (1) and the Type 3 ideal of γ-constacyclic codes for some particular values of p and s. We also present several parameters including the exact symbol-pair distances of MDS constacyclic symbol-pair codes for different values of p and s.

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