MacWilliams identities for the generalized rank weights

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.

💡 Research Summary

This paper investigates the distribution of generalized rank weights (GRWs) of linear codes over finite field extensions and establishes a MacWilliams-type identity linking a code’s GRW distribution to that of its dual. The authors begin by reviewing classical MacWilliams identities for Hamming and rank metrics, noting that while such identities are well‑understood for ordinary rank weights, the generalized case has remained largely unexplored.

In the preliminaries, a linear code C ⊂ F_{q^m}^n of dimension k is considered. For a vector c ∈ C, the rank weight wt_R(c) is defined as the dimension over F_q of the row space of the matrix obtained by expanding each coordinate of c with respect to a fixed F_q‑basis of F_{q^m}. For an r‑dimensional subspace D ⊂ C, the rank weight wt_R(D) is the dimension of the F_q‑span of all row spaces of vectors in D. The r‑th generalized rank weight M_r(C) is then the minimum of wt_R(D) over all D of dimension r. Several equivalent definitions are recalled, and it is shown that the hierarchy M_1(C) < M_2(C) < … < M_k(C) is strictly increasing and satisfies the generalized Singleton bound M_r(C) ≤ n − k + r.

The central object of study is the generalized rank weight distribution A_{r,w}(C) = #{ D ⊂ C | dim_{F_{q^m}} D = r, wt_R(D) = w }, together with the homogeneous enumerator polynomial W_r(C; X, Y) = Σ_{w=0}^n A_{r,w}(C) X^w Y^{n−w}. Basic properties of A_{r,w}(C) are derived, notably that A_{0,0}=1, A_{r,w}=0 for wn, and that A_{1,w} = (q^m−1)·|{c∈C | wt_R(c)=w}|.

Section 3 presents the main MacWilliams identity for GRWs. By working over a generic field extension L/K and using the orthogonal complement with respect to the standard inner product, the authors define for a K‑subspace U ⊂ K^n the set C(U) = {c ∈ C | Rsupp(c) ⊂ U^⊥}. They prove that the number of r‑dimensional subspaces of C whose rank support lies in a given U can be expressed via Gaussian q‑binomial coefficients. Consequently, for any r and w, A_{r,w}(C) = Σ_{i=0}^{n−w} (−1)^i q^{i(i−1)/2} \begin{bmatrix} n−w \ i \end{bmatrix}q A{r,w+i}(C^⊥), which is a direct analogue of the classical MacWilliams relation but now applied to subspace‑level weight enumerators. This identity shows that the GRW distribution of a code determines, and is determined by, that of its dual.

In Section 4 the authors translate the identity into an explicit formula for the enumerator polynomial. Using the q‑analogue of the binomial theorem and the q‑Vandermonde identity, they obtain W_r(C; X, Y) = Σ_{i=0}^{n} (−1)^i q^{i(i−1)/2} \begin{bmatrix} n−i \ r \end{bmatrix}_q X^{i} Y^{n−i}, which can be rewritten as a q‑transform of W_r(C^⊥; X, Y). This closed‑form expression enables efficient computation of the full GRW spectrum from the dual code’s spectrum.

Section 5 focuses on r‑MRD (Maximum Rank Distance) codes, i.e., codes attaining the generalized Singleton bound M_r(C)=n−k+r. The authors prove that for any r‑MRD code the GRW distribution depends only on the parameters (n, k, q, m) and not on the specific code structure. Explicitly, A_{r,w}(C) = \begin{bmatrix} n \ w \end{bmatrix}q Σ{j=0}^{r} (−1)^j q^{j(j−1)/2} \begin{bmatrix} w \ j \end{bmatrix}_q, which coincides with known results for ordinary rank‑weight distributions of MRD codes. Hence the generalized weight spectrum of MRD codes is completely determined by combinatorial quantities.

The paper concludes by emphasizing that the derived MacWilliams identity and enumerator formula fill a notable gap in the theory of rank‑metric codes, providing tools for analyzing error‑correction capabilities, designing optimal network‑coding schemes, and assessing security in cryptographic constructions that rely on rank‑metric properties. Potential future work includes extending these results to generalized sum‑rank weights, non‑linear codes, and codes over rings.