Algebraic Geometry codes in the sum-rank metric
Elena Berardini, Xavier Caruso
TL;DR
The paper introduces linearized Algebraic Geometry codes in the sum-rank metric by leveraging quotients of Ore polynomial rings over function fields of curves. It develops a noncommutative Riemann--Roch theory for these Ore algebras and constructs codes via a multi-evaluation map on Galois covers, providing explicit lower bounds for dimension and minimum distance. In the isotrivial case, the construction recovers linearized Reed--Solomon codes and yields asymptotically good sequences that beat the sum-rank Gilbert--Varshamov bound for suitable parameters, using Drinfeld–Vlăduţ towers. The work extends classical AG-code ideas to a noncommutative setting, offering both theoretical insights and practical avenues for decoding and duality in sum-rank codes.
Abstract
We introduce the first geometric construction of codes in the sum-rank metric, which we called linearized Algebraic Geometry codes, using quotients of the ring of Ore polynomials with coefficients in the function field of an algebraic curve. We study the parameters of these codes and give lower bounds for their dimension and minimum distance. Our codes exhibit quite good parameters, respecting a similar bound to Goppa's bound for Algebraic Geometry codes in the Hamming metric. Furthermore, our construction yields codes asymptotically better than the sum-rank version of the Gilbert-Varshamov bound.