A construction of Abelian non-cyclic orbit codes

Abstract

A constant dimension code consists of a set of k-dimensional subspaces of F n q , where F q is a finite field of q elements. Orbit codes are constant dimension codes which are defined as orbits under the action of a subgroup of the general linear group on the set of all k-dimensional subspaces of F n q. If the acting group is Abelian, we call the corresponding orbit code Abelian orbit code. In this paper we present a construction of an Abelian non-cyclic orbit code for which we compute its cardinality and its minimum subspace distance. Our code is a partial spread and consequently its minimum subspace distance is maximal.

Abelian non-cyclic orbit codes and multishot subspace codes

“…To the best of our knowledge, the first Abelian non-cyclic orbit codes with parameters (n, q(q − 1), 2k, k), [7,Theorem 3], were proposed in [7], satisfying the inequalities…”

“…be the non-Abelian subgroup of the uppertriangular-matrix group (7) P := {A ∈ GL n (F q ) : a ij = 0 for i > j and a ii = 1 for i = 1, 2, ..., n} .…”

“…Recently, new constructions of orbit codes were proposed. For instance, in [7] it is proposed a construction of orbit codes using as generating group a more general Abelian group, whose codes attain the maximum subspace distance. In [3], a wellstructured construction of non-Abelian orbit codes is proposed making use of the semi-direct product of the cyclic group generated by a primitive element of F q n with the cyclic group generated by the Frobenius automorphism.…”

Orbit codes from forms on vector spaces over a finite field

“…By [44,Proposition 3.11], if a code has G as a generating subgroup, then G is a subgroup of the automorphism group of the code. These codes were introduced in [46], and since then they have been further investigated by many authors [8,30,45,40]. It is well known that GL(V ) contains exactly one conjugacy class of cyclic subgroups, acting regularly on V \ {0} and isomorphic to F q n \ {0}, i.e., the Singer groups.…”

“…Typically, the known cyclic codes admit a group of order at most N (q N −1), that is the normalizer of a Singer group, but their size is far behind the theoretical upper bound. In [8] the authors study an abelian non-cyclic group of order q(q − 1) 2 in order to obtain an orbit code with parameters (n, q(q − 1), 2k, k) q .…”