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

在本文中，我们提供了一种递归方法来构造类\(\{k_1,k_2,\ldots ,k_e\}\)和长度n在准伽罗瓦环\(\ mathbb {F}_{2^r}[u]/<u^e>\)来自相同长度n和维度\(k_1+k_2+\cdots +k_{\lceil \frac{e }{2}\rceil }\)在\(\mathbb {F}_{2^r}\)上，反之亦然，其中\(\mathbb {F}_{2^r}\)是有限域order \(2^r,\) \(n \ge 1, \) \(e\ge 2\)是整数，\(\lceil \frac{e}{2}\rceil \)是大于的最小整数大于或等于\(\frac{e}{2},\)和\(k_1,k_2,\ldots ,k_e\)是满足\(k_1 \le n-(k_1+k_2+\cdots +k_e)\)和\(k_i=k_{e-i+2} \)对于\(2 \le i \le e.\)我们进一步应用这种递归方法为环上任意长度的自正交和自对偶码提供显式枚举公式\(\mathbb {F}_ {2^r}[u]/<u^e>\)。借助这些枚举公式，通过在 Magma Computational Algebra 系统中进行计算，我们将环 \(\mathbb {F}_2 上所有长度为 2、3、4、5 的自正交码和自对偶码分类[u]/<u^3>\)并且环上的长度为 2, 3, 4 \(\mathbb {F}_4[u]/<u^2>\)。