Generalized Hamming weights and minimal shifts of Orlik-Terao algebras

2023/11/07 by Ştefan Tohǎneanu, Tohaneanu, Stefan O. · 1 citation
Mathematics · #13P25 (Primary) 14N20 #52C35 #68W30 (Secondary) #Advanced Topics in Algebra #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2311.03688

openalex publication_date 2023/11/07 · openalex created_date 2023/11/09 · openalex updated_date 2026/07/28

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.

Cited by

• The geometry of zonotopal algebras II: Orlik--Terao algebras and Schubert varieties

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