Problem
Research questions and friction points this paper is trying to address.
Etzion-Silberstein conjecture
Ferrers diagram
rank-metric codes
MSRD-constructibility
block Ferrers diagrams
On the Etzion-Silberstein conjecture for block Ferrers diagrams
🤖 AI Summary
This work investigates whether the Etzion–Silberstein bound on the dimension of Ferrers diagram rank-metric codes under block Ferrers diagrams is attainable. To this end, it introduces a novel framework of MSRD-constructibility, constructing optimal codes via maximum sum-rank distance (MSRD) codes placed along the block diagonal and establishing a connection to MDS-constructibility of the associated contraction graph. The authors prove that this construction achieves the dimension upper bound over sufficiently large finite fields and extend the Etzion–Silberstein conjecture to the block-structured setting. Furthermore, they establish MSRD-constructibility for strictly block-monotone and initially block-convex diagrams and, by reduction to block-triangular diagrams, confirm the conjecture in several new cases over arbitrary finite fields.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
MSRD-constructibility
block Ferrers diagrams
rank-metric codes
Etzion–Silberstein conjecture
sum-rank distance
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