An Assmus-Mattson Theorem for Rank Metric Codes

TL;DR

This paper extends the classical Assmus-Mattson theorem to rank metric codes, providing conditions under which codewords of a certain rank form subspace designs, with implications for coding theory and combinatorics.

Contribution

It introduces a rank-metric analog of the Assmus-Mattson theorem, linking rank metric codewords to subspace designs using puncturing, shortening, and MacWilliams identities.

Findings

Established conditions for rank metric codewords to form subspace designs

Connected rank metric code properties with combinatorial design theory

Extended classical design theorems to the rank metric setting

Abstract

A $t$-$(n,d,\lambda)$ design over ${\mathbb F}_q$, or a subspace design, is a collection of $d$-dimensional subspaces of ${\mathbb F}_q^n$, called blocks, with the property that every $t$-dimensional subspace of ${\mathbb F}_q^n$ is contained in the same number $\lambda$ of blocks. A collection of matrices in over ${\mathbb F}_q$ is said to hold a subspace design if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a subspace design.