Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration

TL;DR

This paper introduces iterative lower bounds for the error coefficients of Griesmer optimal linear codes, providing tight bounds for low dimensions and close estimates for higher dimensions, advancing understanding of code optimality.

Contribution

The paper presents new iterative lower bounds for error coefficients of Griesmer optimal codes, improving upon existing bounds and analyzing their tightness across different code dimensions.

Findings

Bounds are tight for binary codes with dimension ≤ 5.

Remaining 5-dimensional codes are characterized and analyzed.

Bounds are within a gap of ≤ 2 from actual error coefficients.

Abstract

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed $5$. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to $2$.