SIAM Journal on Computing, Volume 50, Issue 4, Page 1248-1262, January 2021.
A $k \times n$ matrix over a field is called an MDS (maximum distance separable) matrix if it satisfies the following property: Any $k$ columns of it are linearly independent. Equivalently, its rows span an MDS code. A question arising in coding theory is what zero patterns MDS matrices can have. There is a natural combinatorial condition, called the rectangle condition, which is necessary over any field, and sufficient over exponentially large fields, concretely of size ${n-1 \choose k-1}$. The GM-MDS conjecture of Dau, Song, and Yuen [On the existence of MDS codes over small fields with constrained generator matrices, in 2014 IEEE International Symposium on Information Theory (ISIT), pp. 1787--1791] speculated that whenever the rectangle condition holds, there exist algebraic constructions over much smaller fields of size $n+k-1$, and gave an algebraic conjecture that implies this. In this work, we prove this algebraic conjecture. In an independent and parallel work, Yildiz and Hassibi [Optimum linear codes with support constraints over small fields, in 2018 IEEE Information Theory Workshop (ITW), pp. 1--5] found an alternative proof for the algebraic conjecture.

中文翻译：

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小域上的稀疏 MDS 矩阵：GM-MDS 猜想的证明

SIAM Journal on Computing，第 50 卷，第 4 期，第 1248-1262 页，2021 年 1 月。
一个字段上的 $k \times n$ 矩阵被称为 MDS（最大距离可分离）矩阵，如果它满足以下属性：它的任何 $k$ 列都是线性无关的。同样，它的行跨越一个 MDS 代码。编码理论中出现的一个问题是 MDS 矩阵可以具有哪些零模式。有一个自然的组合条件，称为矩形条件，它在任何场上都是必要的，并且在指数级大场上是足够的，具体来说，大小为 ${n-1 \choose k-1}$。Dau、Song 和 Yuen 的 GM-MDS 猜想 [关于 MDS 码在具有约束生成矩阵的小场上的存在，在 2014 年 IEEE 信息理论国际研讨会 (ISIT)，pp. 1787--1791] 推测，每当矩形条件成立，在大小为 $n+k-1$ 的小得多的域上存在代数构造，并给出了暗示这一点的代数猜想。在这项工作中，我们证明了这个代数猜想。在一项独立且并行的工作中，Yildiz 和 Hassibi [Optimum linear coding with support constraint over small fields, in 2018 IEEE Information Theory Workshop (ITW), pp. 1--5] 找到了代数猜想的替代证明。