Abstract
This paper is concerned with the linear codes over the non-chain ring $R=\mathbb {F}_{2}[v]/\langle v^{4}-v\rangle $ . 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 $R^{n}$ to $\mathbb {F}_{2}^{4n}$ . Finally, several examples are given to illustrate our main results and some open problems are also proposed.
Highlights
Codes over rings are very important in algebraic coding theory and applications to combined coding and modulation
Inspired by the work listed above, we investigate several different weight enumerators of linear codes over a non-chain ring R = F2[v]/ v4 − v, and we give the MacWilliams identity on linear codes over R with respect to these weight
The complete weight enumerator of R−linear code multiplied by equal value may be different, while the symmetric weight enumerator of it is identical
Summary
Codes over rings are very important in algebraic coding theory and applications to combined coding and modulation. Since there have been many different weight enumerators for codes over finite fields and finite rings with respect to the MacWilliams identity. The MacWilliams identities of various weight enumerators for linear codes over non-principal rings were widely studied. Shi et al determined the MacWilliams identities for linear codes with respect to Lee weight enumerator over the rings F2 + vF2 + v2F2 [12] and Fp + vFp [13], respectively. Inspired by the work listed above, we investigate several different weight enumerators of linear codes over a non-chain ring R = F2[v]/ v4 − v , and we give the MacWilliams identity on linear codes over R with respect to these weight.
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