A recursive method for the construction and enumeration of self-orthogonal and self-dual codes over the quasi-Galois ring $\mathbb {F}_{2^r}[u]/<u^e>$

Abstract

In this paper, we provide a recursive method to construct self-orthogonal and self-dual codes of the type $\{k_1,k_2,\ldots ,k_e\}$ and length n over the quasi-Galois ring $\mathbb {F}_{2^r}[u]/<u^e>$ from a self-orthogonal code of the same length n and dimension $k_1+k_2+\cdots +k_{\lceil \frac{e}{2}\rceil }$ over $\mathbb {F}_{2^r}$ and vice versa, where $\mathbb {F}_{2^r}$ is the finite field of order $2^r,$ $n \ge 1,$ $e\ge 2$ are integers, $\lceil \frac{e}{2}\rceil$ is the smallest integer greater than or equal to $\frac{e}{2},$ and $k_1,k_2,\ldots ,k_e$ are non-negative integers satisfying $k_1 \le n-(k_1+k_2+\cdots +k_e)$ and $k_i=k_{e-i+2}$ for $2 \le i \le e.$ We further apply this recursive method to provide explicit enumeration formulae for self-orthogonal and self-dual codes of an arbitrary length over the ring $\mathbb {F}_{2^r}[u]/<u^e>$. With the help of these enumeration formulae and by carrying out computations in the Magma Computational Algebra system, we classify all self-orthogonal and self-dual codes of lengths 2, 3, 4, 5 over the ring $\mathbb {F}_2[u]/<u^3>$ and of lengths 2, 3, 4 over the ring $\mathbb {F}_4[u]/<u^2>$.