Products of Rank-Metric Codes

Abstract. Over the past decade, there has been a renewed interest in studying the component-wise product of codes, leading to new connections between coding theory and additive combinatorics. This framework has also found applications in diverse areas, including secret sharing, multiparty computation, algebraic complexity theory, cryptanalysis, and code-based encryption schemes. While extensively studied for Hamming-metric codes, this approach remains relatively underexplored in the context of rank-metric codes. Inspired by the work of Mirandola and Zemor, in this talk we present bounds on the parameters of the product of Maximum Rank Distance rank-metric codes and show that Gabidulin codes attain these bounds with equality. Additionally, we discuss the implications of these results for decoding algorithms and their potential applications in public key cryptography.

Joint work: The new results in this talk are joint work with A. Couvreur.

Suggested readings: arXiv:1501.06419