General weight spectrum of affine Grassmann codes

Determine the complete weight spectrum of the affine Grassmann code C^A(ℓ,m) for general positive integers ℓ≤m−ℓ, extending the known cases beyond ℓ=2 and the case C^A(3,6).

Background

The paper determines the complete weight distribution of the affine Grassmann code CA(3,6), obtaining 16 distinct weights, and notes that the general affine Grassmann-code problem remains unresolved. On the Grassmann side, weight computation is connected with classifying alternating forms under general linear-group orbits; this classification is known only in limited cases. The affine cell has a smaller symmetry group, producing finer orbit structures, so the Grassmann-code spectrum does not directly determine the affine spectrum.

The authors further explain that, for ℓ=3, weights are not determined solely by a single rank invariant: the arithmetic of the finite field enters through the number of roots of a quadratic polynomial. They identify an inductive strategy based on translating away quadratic terms and specializing one row of the generic matrix, with the cases CA(2,m) and CA(3,6) serving as initial steps.

References

Beyond that, the weight spectrum of C{}(\ell,m) remains open.

— Revisiting the Weight Spectrum of the Affine Grassmann Code $C^{\mathbb{A}}(2,m)$  (2609.10274 - Yadav, 9 Sep 2026) in Section 1, Introduction

The weight spectrum of $C{}(\ell,m)$ for general $\ell$ and $m$ is still open, and Theorem~\ref{thm:full-wd} settles the first case with $\ell\ge3$.

— The Weight Spectrum of the Affine Grassmann Code $C^{\mathbb A}(3,6)$  (2609.08784 - Singh et al., 8 Sep 2026) in Remark 2.14, Section 2.3.3 (the remark labeled \ref{rem:general})