In this paper we investigate partial spreads of H(2n− 1, q2) through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic sem...

In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. The various bounds are then applied to the classes o...

Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed-Solomon codes. We prove that they have maximum sum-rank distance. Such dist...

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 ful...

We present an algorithm that achieves general syndrome decoding of a (n; k; r) linear rank distance code over GF(q m) in O((nr + m) 3 q (m?r)(r?1)) elementary operations. As a consequence, the cryptographic Al schemes Che94, Che96] which rely on this problem are not secure with the proposed parameters. We also derive from our algorithm a bound on the minimal rank distance of a linear code which...

We study properties of rank metric and codes in rank metric over finite fields. We show that perfect codes do not exist. We derive an equivalent of the Varshamov-Gilbert bound in Hamming metric. We study the asymptotic behavior of the minimum rank distance of codes that are on GV. We show that the packing density of maximum rank distance codes is lower bounded by a function depending on the err...

The tensor product of one code endowed with the Hamming metric and rank is analyzed. This gives a which naturally inherits sum-rank metric. Specializing to cyclic skew-cyclic code, resulting turns out belong recently introduced family cyclic-skew-cyclic codes. A group theoretical description these codes given, after investigating semilinear isometries in Finally, generalization Roos Hartmann-Tz...