Affine Cartesian codes with complementary duals

TL;DR

This paper characterizes when generalized affine Cartesian codes, including Reed-Solomon codes, are linear complementary duals (LCD), by analyzing their construction and the scalars involved, contributing to the understanding of dual code properties.

Contribution

It provides a characterization of LCD properties for generalized affine Cartesian codes and their relation to Reed-Solomon codes, expanding the understanding of dual code structures.

Findings

Generalized affine Cartesian codes can be LCD depending on scalar choices.

Reed-Solomon codes are a special case of these codes with LCD properties.

The paper offers a criterion for LCD property in these codes.

Abstract

A linear code $C$ with the property that $C \cap C^{\perp} = \{0 \}$ is said to be a linear complementary dual, or LCD, code. In this paper, we consider generalized affine Cartesian codes which are LCD. Generalized affine Cartesian codes arise naturally as the duals of affine Cartesian codes in the same way that generalized Reed-Solomon codes arise as duals of Reed-Solomon codes. Generalized affine Cartesian codes are evaluation codes constructed by evaluating multivariate polynomials of bounded degree at points in $m$-dimensional Cartesian set over a finite field $K$ and scaling the coordinates. The LCD property depends on the scalars used. Because Reed-Solomon codes are a special case, we obtain a characterization of those generalized Reed-Solomon codes which are LCD along with the more general result for generalized affine Cartesian codes.