On the Etzion-Silberstein conjecture for block Ferrers diagrams

Abstract

Ferrers diagram rank-metric codes are rank-metric codes with prescribed support, and their dimension is bounded from above by the Etzion--Silberstein bound. In this paper, we study this problem for block Ferrers diagrams, namely Ferrers diagrams whose dots are grouped into square blocks of a fixed size. Motivated by the diagonal construction for MDS-constructible Ferrers diagrams, we introduce the notion of MSRD-constructibility, where MDS codes on diagonals are replaced by maximum sum-rank distance (MSRD) codes on block diagonals. We show that MSRD-constructible pairs yield optimal Ferrers diagram rank-metric codes over sufficiently large finite fields. We then relate MSRD-constructibility of a block Ferrers diagram to MDS-constructibility of its contraction, proving an equivalence when the distance is compatible with the block size and giving lifting criteria in the general case. As a consequence, we obtain MSRD-constructibility for strictly block-monotone and initially block-convex diagrams. Finally, we prove a reduction to block triangular diagrams and use it to obtain new arbitrary-field cases of the Etzion--Silberstein conjecture for MSRD-constructible block Ferrers diagrams.