Mass formula for self-dual codes over $\varvec{\mathbb {F}}_q+u\varvec{\mathbb {F}}_q+u^2\varvec{\mathbb {F}}_q$

Abstract

In this paper, we establish a mass formula for self-dual codes over the finite chain ring $\mathbb {F}_q+u\mathbb {F}_q+u^2\mathbb {F}_q$, where $\mathbb {F}_q$ is the finite field of order q and $u^3=0$. We also give a classification of self-dual codes over $\mathbb {F}_q+u\mathbb {F}_q+u^2\mathbb {F}_q$, for $q=2,3,4,5,7,8,9$, with lengths 2 and 4.