Linear Codes Over a Non-Chain Ring and the MacWilliams Identities

Abstract

This paper is concerned with the linear codes over the non-chain ring R=F2[v]/〈v4-v〉. First, several weight enumerators over R are defined. Then the MacWilliams identity is obtained, which can establish an important relation respect to the complete weight enumerators. Meanwhile, the symmetric weight enumerators between linear code and its dual over R are established by the Gray map from Rn to F24n. Finally, several examples are given to illustrate our main results and some open problems are also proposed.

Keywords

• Generator matrix
• linear code
• MacWilliams identity
• weight enumerator