MacWilliams identities for the generalized rank weights
Julien Molina
TL;DR
This work addresses the generalized rank weight distribution of $[n,k]$ codes over $\mathbb{F}_{q^m}$ under the rank metric by establishing a MacWilliams-type identity relating a code to its dual and by deriving a closed-form generalized rank weight enumerator $W_\mathcal{C}^r(X,Y)$ in terms of auxiliary counts $B_t^r(\mathcal{C})$ and Gaussian binomial coefficients. It then specializes to MRD codes, proving that the GRW distribution depends only on the parameters $(n,m,k)$, with explicit expressions for $A_w^r(\mathcal{C})$ and the dual MRD bounds $M_r(\mathcal{C})=n-k+r$ and $M_r(\mathcal{C}^\perp)=k+1$, and confirming this dependence through concrete examples. The results provide a duality framework and practical formulas for analyzing rank-metric codes, including MRD codes, with potential impact on network coding and related applications. Overall, the paper advances the understanding of duality in the rank-metric setting and offers explicit tools for computing generalized weight distributions.
Abstract
We study the generalized rank weight distribution of a linear code. First, we provide a MacWilliams-type identity which relates the distributions of a code and its dual. Then, we give a formula for the enumerator polynomial. Finally, we explicitly compute the distribution of an MRD code.