MacDonald codes over the ring $${\mathbb {F}}_{p}+v{\mathbb {F}}_{p}+v^2{\mathbb {F}}_{p}$$

Abstract

In this paper, we consider MacDonald codes over the finite non-chain ring $${\mathbb {F}}_p+v{\mathbb {F}}_p+v^2{\mathbb {F}}_p$$ and their applications in constructing secret sharing schemes and association schemes, where p is an odd prime and $$v^3=v$$ . We give some structural properties of MacDonald codes first. Then, we study the weight enumerators of torsion codes of these MacDonald codes. As some applications, constructing secret sharing schemes and association schemes is also investigated.