A geometric approach to rank metric codes and a classification of constant weight codes

Abstract

In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes. We give a complete classification of constant rank weight code and we give their generalized rank weights. *

The geometry of one-weight codes in the sum-rank metric

“…This connection has been intensively used to get classification results and intriguing constructions in both the areas of coding theory and finite geometry. Recently, in [38,41] it has been shown that equivalence classes of nondegenerate rank-metric codes are in one-to-one correspondence with equivalence classes of q-systems, where the latter constitute the q-analogue of projective systems; see also [1]. At this point, it is natural to ask whether it is possible to construct geometric objects able to capture the structure of sum-rank metric codes which generalize both projective systems and q-systems.…”

“…More recently a similar geometric approach was used for codes in the rank metric. A first intuition was given in [40], while a full correspondence was provided in [41] and in [38]: kdimensional rank-metric codes over an extension field F q m of F q correspond to q-systems, which are special F q -subspaces of F k q m . This correspondence allowed us to give a complete characterization of nondegenerate F q m -linear one-weight codes in the rank metric; see [38,Theorem 12], [1,Proposition 3.16].…”

Evasive subspaces

“…Recently, in [24] Randrianarisoa introduced q-systems. A q-system U over q n  with parameters m r d [ , , ] is an m-dimensional q  -subspace generating over q n  a r-dimensional q n  -vector space V , where…”

“…]-rank metric codes, cf. [24,Theorem 2]. In [10] the authors investigated the following subspace analogue of caps of kind h: for h r…”

Multi-Sidon spaces over finite fields

“…Let G be a matrix in F r× rn 2 q m whose columns are a basis of U . As a consequence of Theorem 7.16 and the connection between linear rank metric codes and q-systems (see [28] and also [1]), the linear rank metric codes having as a generator matrix G have exactly three nonzero weights, which are n − r, n − 1, n. In particular, they are examples (r − 1)-almost MRD codes, see [12]. Moreover, using [1, Theorem 4.8], we can from L U we can also construct linear Hamming metric codes with only three weight and for which we can completely establish its weight distribution, as already done for some classes of linear sets (see also [25,33]).…”