Negacyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$ of oddly even length and their Gray images

Let R = Z4[v]/〈v 2 + 2v〉 = Z4 + vZ4 (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 Z4-linear Gray map from R onto Z 2 4 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 · (4+5 ·2+9) 2 p −2 p negacyclic codes over R of length 2Mp and all 3 ·(4 +5 ·2+9) 2p−1−1 p self-dual codes among them are presented precisely, where Mp = 2 p − 1 is a Mersenne prime. Finally, 36 new and good self-dual 2-quasi-twisted linear codes over Z4 with basic parameters (28, 2 , dL = 8, dE = 12) and of type 2 4 and basic parameters (28, 2, dL = 6, dE = 12) and of type 2 4 which are Gray images of self-dual negacyclic codes over R of length 14 are listed.