Negacyclic codes over the local ring Z_4[v]/〈 v^2+2v〉 of oddly even length and their Gray images
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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 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+9)^2^p-1-1/p 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 2^144^7 and basic parameters (28,2^28, d_L=6,d_E=12) and of type 2^164^6 which are Gray images of self-dual negacyclic codes over R of length 14 are listed.
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