Decreasing norm-trace codes

TL;DR

This paper introduces decreasing norm-trace codes, analyzing their parameters, duals, and conditions for self-orthogonality or self-duality, and presents an efficient repair scheme for single erasures.

Contribution

It defines decreasing norm-trace codes, determines their parameters using Gröbner bases, and develops a linear repair scheme surpassing previous methods for certain codes.

Findings

Determined length, dimension, and minimum distance of the codes.

Derived conditions for self-orthogonality and self-duality.

Provided an improved repair scheme for single erasures.

Abstract

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gr\"obner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin, Luo, and Xing (IEEE Transactions on Information Theory {\bf 64} (2), 900-908, 2018) when applied to the one-point AG codes over the extended norm-trace curve.