Ferrers diagram rank-metric (FDRM) code [1]

Description

Linear rank-metric code whose codeword matrices vanish outside a fixed Ferrers diagram \(\mathcal{F}\) [1]. The diagram is an \(m\times n\) array of right-justified dots with non-increasing row lengths, i.e., the free entries of a reduced row echelon form with fixed pivots [1]. Inserting each codeword into the free entries of an echelon form with these pivots lifts the code to subspaces whose subspace distance is twice the rank distance, the building blocks of the multilevel construction of constant-dimension codes [1].

An FDRM code on \(\mathcal{F}\) with dimension \(k\) and minimum rank distance \(\delta\) is denoted \([\mathcal{F},k,\delta]_q\). The diagram is written \([\gamma_0,\ldots,\gamma_{n-1}]\) by its column lengths. A standard construction restricts the \(i\)th information symbol of a systematic MRD code with generator matrix \((I_\kappa|A)\), \(\kappa=n-\delta+1\), to the span of the first \(\gamma_i\) elements of an \(\mathbb{F}_q\)-basis of \(\mathbb{F}_{q^m}\) [2]. This confines the first \(\kappa\) columns of each codeword to the diagram while keeping rank distance at least \(\delta\), and the subcode is an FDRM code when the rightmost columns are long enough [2,3]. Not every optimal FDRM code is a subcode of an MRD code [3].

Optimal codes on additional diagram families are obtained as subcodes of MRD codes with explicit systematic generator matrices, from systematic MRD codes extended by columns that copy their leading information symbols, and by combining codes on disjoint subdiagrams [2,4][5; Thms. 1-3 and Secs. 4 and 5].