Two-weight linear codes are linear codes in which any nonzero codeword can have only two possible distinct weights. Those in the Hamming metric have proven to be very interesting for their connections with authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we characterize two-weight codes in the rank metric, answering a recent question posed by Pratihar and Randrianarisoa.

Abstract. In this paper, we investigate the problem of determining the number of codewords with maximum rank weight in a linear rank-metric code. We also characterize the codes that reach the maximum or the minimum value for this parameter.

A subspace of matrices in${\mathbb F}_{q^{e}}^{m\times n}$can be naturally embedded as a subspace of matrices in${\mathbb F}_{q}^{em\times en}$with the property that the rank of any of its matrix is a multiple of$e$. It is quite natural to ask whether or not all subspaces of matrices with such a property arise from a subspace of matrices over a larger field. In this paper we explore this question, which corresponds to studying divisible codes in the rank metric. We determine some cases for which this question holds true, and describe counterexamples by constructing subspaces with this property which do not arise from a subspace of matrices over a larger field.