Abstract
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.
Highlights
Codes over finite rings has been an interesting topic in algebraic coding theory since the discovery of codes over Z4, see [4]
We first discuss about the form of maximal ideals and characterization of automorphisms for the ring Bk
We give a characterization for equivalence of codes over Bk and Euclidean self-dual codes
Summary
Codes over finite rings has been an interesting topic in algebraic coding theory since the discovery of codes over Z4, see [4]. Vk], where vi2 = vi, for 1 ≤ i ≤ k, because it has two Gray maps which relate codes over such ring and binary codes, see [2] This ring has non-trivial automorphisms which can be used to define skew-cyclic codes, for example in [1], skew-cyclic codes over the ring A1 = F2 + vF2, where v2 = v, which give some optimal Euclidean and Hermitian self-dual codes. We study its maximal ideals, automorphisms, equivalence codes, and Euclidean self-dual codes over these rings, including the generators for its invariant ring.
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