Explicit and asymptotically good constructions of Algebraic Geometry codes in the sum-rank metric
2026-06-08 • Information Theory
Information Theory
AI summaryⓘ
The authors study a type of error-correcting codes called Linearized Algebraic Geometry codes, which are similar to traditional Algebraic Geometry codes but work in a different mathematical setting called the sum-rank metric. These newer codes were introduced recently using advanced polynomial rings related to algebraic function fields. The authors build on prior work by providing clear, optimal, and scalable ways to construct these codes. Their results help better understand the structure and potential of these sum-rank metric codes.
Algebraic Geometry codesLinear codesSum-rank metricAlgebraic function fieldsOre polynomialsError-correcting codesCoding theoryAsymptotic constructions
Authors
Peter Beelen, Elena Berardini, Anina Gruica, Maria Montanucci
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.