The Support-Distribution Conditions for MSRD Codes Reduce to a Single Inequality

Abstract

Byrne, Gluesing-Luerssen, and Ravagnani determined the number of codewords with a prescribed nonzero support in a linear maximum sum-rank distance code whose blocks have a common number of columns. Their formula depends only on the code parameters and the support dimensions. Evaluating it for arbitrary parameters gives a family of necessary nonnegativity conditions for existence. We prove their conjecture that this family is equivalent to the condition at the vector of total dimension $d+1$ obtained by filling the longest blocks first, where $d$ is the distance parameter. The resulting explicit criterion applies over every finite field $\mathbb F_q$, for all row numbers $n_i\leqslant m$ and $1\leqslant d<N$, where $m$ is the common number of columns and $N$ is the total number of rows. It also covers the boundary cases in which this vector fills each of its nonzero blocks completely, not explicitly included in the conjecture's displayed parameter range. The proof uses a generating function and an exchange identity to reduce the problem to vectors obtained by filling the longest blocks first. We then estimate alternating sums of elementary symmetric polynomials, with a separate argument for $q=2$. The dual conditions admit the same reduction. For codes that exist, the joint condition follows from sphere-packing bounds for the code and its dual. An infinite family of parameters satisfies the joint condition but admits no such linear code.