A recursive method for the construction and enumeration of self-orthogonal and self-dual codes over the quasi-Galois ring F2r[u]/< ue >

In this paper, we provide a recursive method to construct self-orthogonal and self-dual codes of the type {k(1), k(2), ... , k(e)} and length n over the quasi-Galois ring F(2)r[u]/ < u(e) > from a self-orthogonal code of the same length n and dimension k(1) + k(2) + middot middot middot + k([e/2])over F(2)r and vice versa, where F2r is the finite field of order 2r, n > 1, e > 2 are integers, [e2] is the smallest integer greater than or equal to e/2, and k(1), k(2), . . . , k([e/2])are non-negative integers satisfying k(1) <= n - (k(1) + k(2) + middot middot middot + k(e)) and k(i) = k(e-i+2) for 2 <= i <= 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 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 F-2[u]/ < u(3) > and of lengths 2, 3, 4 over the ring F-4[u]/ < u(2) >.