关于分块费勒斯图的埃齐翁 - 西尔伯斯坦猜想
On the Etzion-Silberstein conjecture for block Ferrers diagrams
AI总结：
研究分块费勒斯图的埃齐翁 - 西尔伯斯坦猜想，引入MSRD可构造性概念，证明其在有限域上能产生最优码，关联其与收缩图的MDS可构造性，得出相关图的MSRD可构造性，还证明向分块三角图的约简及猜想新情形。
摘要：
费勒斯图秩度量码是具有规定支撑的秩度量码，其维度受埃齐翁 - 西尔伯斯坦界的上限限制。本文针对分块费勒斯图研究此问题，即点被分组为固定大小方块的费勒斯图。受可构造MDS费勒斯图的对角线构造启发，引入MSRD可构造性概念，用块对角线上的最大和秩距离（MSRD）码取代对角线上的MDS码。证明了在足够大的有限域上，MSRD可构造对产生最优费勒斯图秩度量码。还将分块费勒斯图的MSRD可构造性与其收缩的MDS可构造性相关联，得出严格分块单调和初始分块凸图的MSRD可构造性。最后证明了向分块三角图的约简，并用于获得MSRD可构造分块费勒斯图的埃齐翁 - 西尔伯斯坦猜想的新的任意域情形。
英文摘要：
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.
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