Negacyclic codes over the local ring Z4[v]/〈v2+2v〉 of oddly even length and their Gray images

Let R=Z4[v]/〈v2+2v〉=Z4+vZ4 (v2=2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z4-linear Gray map from R onto Z42 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 length 2n are presented. Moreover, all 23⋅(4p+5⋅2p+9)2p−2p negacyclic codes over R of length 2Mp and all 3⋅(4p+5⋅2p+9)2p−1−1p self-dual codes among them are presented precisely, where Mp=2p−1 is a Mersenne prime. Finally, 36 new and good self-dual 2-quasi-twisted linear codes over Z4 with basic parameters (28,228,dL=8,dE=12) and of type 21447 and basic parameters (28,228,dL=6,dE=12) and of type 21646 which are Gray images of self-dual negacyclic codes over R of length 14 are listed.