# 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.

### References

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.

J. Gao, Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform. 31(3–4) (2013) 337–342.

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.

W. Huffman, V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003.

Irwansyah, I. Muchtadi–Alamsyah, A. Muchlis, A. Barra, D. Suprijanto, Construction of thetaθ–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.

J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.