Codes over an infinite family of algebras

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.

Keywords

• Gray map,Equivalence of codes,Euclidean self-dual,Hamming weight enumerator,MacWilliams relation,Invariant ring

References

1. [1] T. Abualrub, N. Aydin, P. Seneviratne, On $\theta-$cyclic codes over F2 + vF2; Australas. J. Combin. 54 (2012) 115–126.
2. [2] Y. Cengellenmis, A. Dertli, S. T. Dougherty, Codes over an infinite family of rings with a Gray map, Des. Codes Cryptogr. 72(3) (2014) 559–580.
3. [3] J. Gao, Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform. 31(3–4) (2013) 337–342.
4. [4] A.R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, P. Sole, The Z4–linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301–319.
5. [5] W. Huffman, V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003.
6. [6] Irwansyah, I. Muchtadi–Alamsyah, A. Muchlis, A. Barra, D. Suprijanto, Construction of $\theta$–cyclic codes over an algebra of order 4, Proceeding of the Third International Conference on Computation for Science and Technology (ICCST–3), Atlantis Press, 2015.
7. [7] J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.