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

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 \(\mathbb {F}_4[u]/<u^2>\).