Enumeration of Linear Codes by Applying Methods from Algebraic Combinatorics

It is demonstrated how classes of linear (n, k)-codes can be enumerated using cycle index polynomials and other methods from algebraic combinatorics. Some results of joined work [9] with Prof. Kerber from the University of Bayreuth on the enumeration of linear codes over GF (q) are presented. Furthermore I will give an introduction to enumeration under finite group actions. At first let me draw your attention to the enumeration of linear codes. Let p be a prime and let q be a power of p then GF (q) denotes the finite field of q elements. A linear (n, k)-code over the Galois field GF (q) is a k-dimensional subspace of the vector space GF (q). As usual codewords will be written as rows x = (x1, . . . , xn). 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 · Γ | x ∈ GF (q)}. The Hamming distance d(x, y) := |{i ∈ n xi 6= yi}| is a metric on GF (q). (The set of integers from 1 to n will be indicated as n.) The minimal distance d(C) of a code C is given by d(C) := min (x,y)∈C2, x6=y d(x, y). It can be used to express the quality of a code. A maximum likelihood decoding algorithm for instance corrects (d − 1)/2 errors and detects d − 1 errors, when d is the minimal distance of the code. In coding theory 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. This means there is a linear isomorphism φ:C1 → C2 such that d(x, y) = d(φ(x), φ(y)) for all x, y ∈ C1. It can be shown that such a linear isometry between two linear (n, k)codes can always be extended to an isometry of GF (q). (See [10].) Let’s investigate the structure of the group of all linear isometries of GF (q). It is enough to consider ∗Supported by a Forschungsstipendium of the University of Graz and by the Fonds zur Förderung der wissenschaftlichen Forschung P10189 PHY. 2 Harald Fripertinger an isometry φ acting on the standard basis {e1, . . . , en} where ei = (δi,j)j∈n. Since φ is a homomorphism, φ(ei) = ∑n j=1 αi,jej, where αi,j ∈ GF (q). Since φ is an isometry, for each i there is exactly one j such that αi,j 6= 0. This defines a function π:n → n such that αi,π(i) 6= 0 and φ(ei) = αi,π(i)eπ(i). Since φ is an isomorphism, π must be a permutation, i.e. π ∈ Sn. Let’s define ψ(i) := αi,π(i), then ψ is a mapping from n to GF (q)∗ (where GF (q)∗ denotes the multiplicative group of the Galois field), and φ can be identified with the pair (ψ, π). Then the group of all isometries corresponds to the wreath product GF (q)∗ o Sn = { (ψ, π) ψ ∈ GF (q)∗, π ∈ Sn} , which is the semidirect product of GF (q)∗ and Sn, where the multiplication is given by (ψ, π)(ψ′, π′) = (ψψ′ π, ππ ′), where ψψ′ π(i) = ψ(i)ψ ′ π(i) and ψ ′ π(i) = ψ ′(π−1i). (From now on GF (q) will be identified with GF (q), the set of all mappings from n to GF (q).) The complete monomial group GF (q)∗ o Sn of degree n over GF (q)∗ acts on GF (q) in the form of the exponentiation, i.e. GF (q)∗ o Sn×GF (q) → GF (q), ((ψ, π), (x1, . . . , xn)) 7→ (ψ(1)xπ−11, . . . , ψ(n)xπ−1n). For computing the number of isometry classes of linear (n, k)-codes we use methods from algebraic combinatorics, which shall be described now. Let me start with the basic concept of a finite group action. (More details can be found in [11].) Let G denote a multiplicative finite group and X a nonempty set. A finite group action GX of G on X is described by a mapping G×X → X, (g, x) 7→ gx, such that g(g′x) = (gg′)x, and 1x = x. In other words, there is a group homomorphism δ from G into the symmetric group SX on X (i.e. the set of all permutations of X) which is called a permutation representation of G on X: δ:G→ SX , g 7→ δ(g) =: ḡ, where ḡ(x) := gx. A group action GX defines the following equivalence relation on X: x ∼G x′ iff ∃g ∈ G: x′ = gx. The equivalence classes are called orbits , and the orbit of x ∈ X will be indicated as G(x) : = {gx g ∈ G}. The set of all orbits will be denoted by G\\X := {G(x) x ∈ X} . Enumeration of Linear Codes 3 For each x ∈ X the stabilizer Gx of x Gx : = {g gx = x} is a subgroup of G. The stabilizer of y = gx is given by Gy = gGxg −1, so the stabilizers of all elements in the orbit of x form the conjugacy class of the subgroup Gx of G. The mapping G(x)→ G/Gx, gx 7→ gGx is a bijection. So we conclude that |G(x)| = |G|/|Gx|. Finally the set of all fixed points of g ∈ G is denoted by Xg : = {x gx = x}. The main lemma in the theory of enumeration under finite group actions is the so called Lemma of Cauchy Frobenius. It says that the number of orbits of a finite group G acting on a finite set X is equal to the average number of fixed points: