Generalized Hamming weights and minimal shifts of Orlik-Terao algebras
Stefan O. Tohaneanu
TL;DR
The paper establishes a direct link between coding-theoretic parameters and homological invariants of the Orlik-Terao algebra: the minimum distance $d$ of a linear code satisfies $d=\alpha(IOT(\mathcal{C}^{\perp}))+1$, where $\alpha$ is the initial degree of the OT ideal. It then analyzes the second generalized Hamming weight $d_2(\mathcal{C})$, proving the bounds $t_2(IOT(\mathcal{C}^{\perp}))+1 \le d_2(\mathcal{C}) \le t_2(IOT(\mathcal{C}^{\perp}))+2$, with $t_2(I)$ the minimal shift in the second syzygy of the OT ideal, and providing examples where these bounds are tight. The study leverages the interplay between hyperplane arrangements, parity-check matrices, and the OT algebra, highlighting both the strong connections to Stanley-Reisner/broken-circuit complexes and the subtle, non-monotone behavior of OT’s syzygies. Overall, the work offers a homological perspective on generalized Hamming weights, clarifying what can and cannot be read off OT invariants alone and suggesting directions for higher-weight analysis beyond the second weight.
Abstract
In this note we show that the minimum distance of a linear code equals one plus the smallest shift in the first step of the minimal graded free resolution of the Orlik-Terao algebra (i.e., the initial degree of the Orlik-Tearo ideal) constructed from any parity-check matrix of the linear code. We move forward with this connection and we prove that the second generalized Hamming weight equals one or two plus the smallest shift at second step in the minimal graded free resolution of the same algebra. Via a couple of examples we show that this ambivalence is the best result one can get if one uses Orlik-Terao algebras to characterize the second generalized Hamming weight.