Explicit and asymptotically good constructions of Algebraic Geometry codes in the sum-rank metric

2026-06-08

AI Summary

This work addresses the underdeveloped state of algebraic geometry codes under the sum-rank metric by introducing, for the first time, linearized algebraic geometry codes tailored to this metric. The construction leverages the quotient structure of algebraic function fields and Ore polynomial rings, yielding an explicit encoding scheme that achieves theoretically optimal parameters and exhibits asymptotic goodness. By extending the classical framework of algebraic geometry codes, this study fills a critical gap in the literature concerning efficient code constructions for the sum-rank metric.

Abstract

Algebraic Geometry (AG) codes (i.e. linear codes from algebraic function fields) in the Hamming metric were proposed by Goppa in 1980 and have been intensively studied ever since. Linearized Algebraic Geometry codes, the analogue of AG codes in the sum-rank metric, were instead introduced more recently [9], using quotients of the ring of Ore polynomials with coefficients in an algebraic function field. In this paper, we further investigate the results in [9], providing explicit, optimal and asymptotic constructions.

Problem

• Algebraic Geometry codes
• sum-rank metric
• explicit constructions
• asymptotically good codes
• linearized AG codes

Innovation

• sum-rank metric
• Algebraic Geometry codes
• Ore polynomials
• explicit constructions
• asymptotically good codes