Codes with locality from cyclic extensions of Deligne-Lusztig curves

TL;DR

This paper constructs locally recoverable codes from cyclic extensions of Deligne-Lusztig curves, achieving smaller locality than existing codes, and explores three different construction methods for various scenarios.

Contribution

It introduces new locally recoverable codes from cyclic extensions of Deligne-Lusztig curves, expanding the class of algebraic-geometry codes with improved locality properties.

Findings

Codes have smaller locality than typical in literature

Three construction methods for different situations

Codes with availability using fiber products and code products

Abstract

Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by accessing only a few others which form what is known as a recovery set. If every symbol has at least two disjoint recovery sets, the code is said to have availability. Three constructions are described, as each best fits a particular situation. The first employs the original construction of locally recoverable codes from curves by Tamo and Barg. The second yields codes with availability by appealing to the use of fiber products as described by Haymaker, Malmskog, and Matthews, while the third accomplishes availability by taking products of codes themselves. We see that cyclic extensions of the Deligne-Lusztig curves provide codes with smaller locality than those typically found in the literature.