Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric

Abstract

This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the parameters of a minimal rank-metric codes, a general existence result based on a combinatorial argument, and an explicit code construction for some parameter sets that uses the notion of a scattered linear set. Throughout the paper we also show and comment on curious analogies/divergences between the theories of error-correcting codes in the rank and in the Hamming metric.

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.…”

“…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].…”

Multi-Sidon spaces over finite fields

“…Indeed, consider the vector u whose i-th entry is x, j-th entry is y and all the other entries are zero and the vector u ′ whose i-th entry is x ′ , j-th entry is y ′ and all the other entries are zero, whereas x, y, x ′ , y ′ are nonzero. By (1) we obtain u = λu ′ , a contradiction to a). Now, suppose that ii) holds.…”

“…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]).…”

“…[29, Lemma 36] Let U, V be two distinct F q -subspaces of dimension k of F q n . Then the following conditions are equivalent (1) dim Fq (U ∩ αV ) ≤ 1, for any α ∈ F q n ;…”