Codes over an infinite family of algebras
Keywords:
Gray map, Equivalence of codes, Euclidean self-dual, Hamming weight enumerator, MacWilliams relation, Invariant ringAbstract
In this paper, we will show some properties of codes over the ring $B_k=\mathbb{F}_p[v_1,\dots,v_k]/(v_i^2=v_i,\forall i=1,\dots,k).$ These rings, form a family of commutative algebras over finite field $\mathbb{F}_p$. We first discuss about the form of maximal ideals and characterization of automorphisms for the ring $B_k$. Then, we define certain Gray map which can be used to give a connection between codes over $B_k$ and codes over $\mathbb{F}_p$. Using the previous connection, we give a characterization for equivalence of codes over $B_k$ and Euclidean self-dual codes. Furthermore, we give generators for invariant ring of Euclidean self-dual codes over $B_k$ through MacWilliams relation of Hamming weight enumerator for such codes.
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References
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