Enumeration of isometry-classes of linear (n, k)-codes over GF (q) in SYMMETRICA

Isometry classes of linear codes can be described as orbits of gener-ator matrices, as it was shown by Slepian. The author demonstrates how they can be enumerated using cycle index polynomials and the tools already incorporated in SYMMETRICA, a computer algebra package devoted to representation theory and combinatorics of sym-metric groups and of related classes of groups. 1 Isometry classes of linear codes A linear (n, k)-code over the Galois field GF (q) is a k-dimensional subspace of the vector space Y X: = GF (q)n, where n denotes the set {0, 1,..., n − 1}. As usual codewords will be written as rows x = (x0,..., xn−1). A k × n-matrix Γ over GF (q) is called a generator matrix of the linear (n, k)-code C, if and only if the rows of Γ form a basis of C, so that C = {x · Γx ∈ GF (q)k}. Two linear (n, k)-codes C1, C2 are called equivalent, if and only if there is an isometry (with respect to the Hamming metric) which maps C1 onto C2. Using the notion of finite group actions (see [8]) one can easily express equivalence of codes in terms of the wreath product action: C1 and C2 are equivalent, if and only if there exist (ψ, pi) ∈ GF (q) ∗ o Sn ={ (ψ, pi) ψ ∈ (GF (q)∗)n, pi ∈ Sn} (where GF (q) ∗ denotes the multiplicative group of the Galois field) such that (ψ, pi)(C1) = C2. The complete monomial group GF (q) ∗ oSn of degree n over GF (q) ∗ acts on GF (q)n by the following definition: GF (q) ∗ o Sn ×GF (q)n → GF (q)n (ψ, pi) ((xi)i∈n) = ψ(i)xpi−1(i) i∈n