On (1 − u 2)-cyclic and cyclic codes over IFp + uIFp + u 2 IFp

Abstract

It is extended a result of [5] to codes over the commutative ring IFp + uIFp + u 2 IFp where u 3 = 0. A new Gray map between codes over IFp + uIFp + u 2 IFp and IFp is defined. It is proved that the Gray image of the linear (1 − u 2)-cyclic code over the commutative ring IFp + uIFp + u 2 IFp of length n is a distance-invariant quasi-cyclic code of index p and length p 2 n over IFp . And it is proved that if (n, p) = 1, then every code of length p 2 n over IFp which is the Gray image of a linear cyclic code of length n over IFp + uIFp + u 2 IFp is permutation equivalent to a quasi-cyclic code of index p.