Isometries for rank distance and permutation group of Gabidulin codes

The rank distance was introduced in 1985 by E. Gabidulin [1]. He determined a lower bound for the minimum rank distance of a code. Moreover, he constructed a class of codes which meet this bound: the so-called Gabidulin codes. In this paper, we first characterize the linear isometries for the rank distance. Then we determine the isometry group and the permutation group of Gabidulin codes of full length (i.e. the length is equal to the degree of the field extension). I. Isometries for rank distance Let K = GF (q) be an extension of degree m of the finite field GF (q). Let E = K be the vector space of dimension n over K. Definition 1 For a ∈ E, a = (a1, . . . , an), the rank rk(a) of a is the dimension of the GF (q)-vector space generated by {a1, . . . , an}. Let a and b be two elements of E. The relation dr(a, b) = rk(a− b) defines a distance over E. Following this definition, it is natural to define the minimum rank distance dr of a code C. Moreover, if dh denotes the classical Hamming distance, then for all a, b in E, the rank distance satisfies the inequality dr(a, b) ≤ dh(a, b). The group of linear isometries for the classical Hamming distance is well-known: it is the monomial group of n × n matrices over K with one and only one non-zero element on each row and each column [2]. This group is generated by the permutations of the support and the scalar multiplications by invertible elements on each coordinate. In this section, we characterize the linear transformations that are isometries for the rank distance. Definition 2 An isometry for the rank distance is a K-linear automorphism f of E which preserves the rank of the elements of E, i.e. rk(a) = rk(f(a)) for all a in E. Let Iso(E) be the group of isometries for the rank distance. The following facts are very easy to check: • The scalar multiplications hλ : a = (a1, . . . , an) %→ λa = (λa1, . . . ,λan), λ ∈ GF (qm)∗ are isometries for the rank distance. • For all M ∈ GL(n, q), the K-linear endomorphism fM of E defined by a %→ aM is an isometry for the rank distance. The following theorem characterizes the isometries for the rank distance. Theorem 1 The isometry group Iso(E) for the rank distance is generated by the scalar multiplications hλ, λ ∈ GF (qm)∗ and the linear group GL(n, q). This group is isomorphic to the product group (GF (qm)∗/GF (q)∗)×GL(n, q). Proof : As noticed previously, the scalar multiplications and the transformations associated to n× n invertible matrices with coefficients in GF (q) are isometries for the rank distance. Let f ∈ GL(n, q) be an invertible K-linear transformation, and M be its associated matrix in the canonical basis. The i-th row of M is the image of ei by f . Suppose that f is an isometry for the rank distance. The rank of each row must be one. Moreover, eventually using a scalar multiplication, it is possible to suppose that the elements of the first row are in GF (q), i.e. f(e1) ∈ GF (q). Let i ∈ {2, . . . , n}. Following the preceding remarks, there exists a μ ∈ K∗ such that μf(ei) is in GF (q) . This is μfi,j ∈ GF (q) for all j = 1, . . . , n. Let c be e1+ei. The rank of c is 1. Its image is f(c) = (f1,1+fi,1, f1,2+fi,2, . . . , f1,n+fi,n) and must be of rank 1. There exists at least one non-zero coordinate, for example the first. Set ν = f1,1 + fi,1 '= 0. Since the rank of f(c) is 1, for a fixed j there exists a s ∈ GF (q) such that f1,j + fi,j = sν, i.e. f1,j + fi,j = s(f1,1 + fi,1). From this fact, we deduce f1,j − sf1,1 = −fi,j + sfi,1. Set t = f1,j − sf1,1. This is an element of GF (q) ∩ μGF (q). Then either μ is in GF (q) and the elements of the i-th row are in GF (q), or t = 0, that implies f1,j = sf1,1 for all j: the i-th row is deduced from the first by multiplication by s. This is not possible, since the matrix M is invertible. This proves the fact that al the fi,j are in GF (q) and f is in GL(n, q). To complete the proof, we first remark that the scalar multiplications hλ commute with all the linear transformations. Moreover, the intersection of the linear group GL(n, q) and the group of scalar multiplications is the subgroup of scalar multiplications for which λ is in GF (q)∗. This implies that Iso(E) is isomorphic to the direct product (K∗/GF (q)∗)× GL(n, q). !