Ştefan O. I. Tohǎneanu: Generalized Hamming weights and minimal shifts of Orlik-Terao algebras, 91-106

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-Terao 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.

Key Words: Generalized Hamming weights, graded Betti numbers, Orlik-Terao algebra, initial degree.

2020 Mathematics Subject Classification: Primary 13P25; Secondary 14N20, 52C35, 68W30.

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