Negacyclic codes over the local ring Z_4[v]/(v~2 + 2v) of oddly even length and their Gray images

摘要

Let R = Z(4)[v]/ v(2) + 2v = Z(4) + vZ(4) (v(2) = 2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z(4)-linear Gray map from R onto Z(4)(2) which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over R of length 2n are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over R of 2 length 2n are presented. Moreover, all 23 . (4(P) + 5 . 2(P) + 9)2(p)-2/p negacyclic codes over R of length 2M(p) and all 3 . (4(P) + 5 . 2(P) + 2(p-1)-1 /9) self-dual codes among them are presented precisely, where M-P = 2(P) - 1 is a Mersenne prime. Finally, 36 new and good self-dual 2 -quasi-twisted linear codes over Z(4) with basic parameters (28, 2(28), d(L) = 8, d(E) = 12) and of type 2(14)4(7) and basic parameters (28, 2(28), d(L) = 6, d(E) = 12) and of type 21848 which are Gray images of self-dual negacyclic codes over R of length 14 are listed. (C) 2018 Elsevier Inc. All rights reserved.