Optimal Few-GHW Linear Codes and Their Subcode Support Weight Distributions

TL;DR

This paper constructs and analyzes a new class of linear codes that meet the Griesmer bound for certain generalized Hamming weights, have few subcode support weights, and generalize known distance-optimal few-weight codes.

Contribution

It introduces a new family of linear codes that achieve the Griesmer bound for specific generalized Hamming weights and have simplified subcode support weight distributions.

Findings

Codes meet the Griesmer bound for r-generalized Hamming weights.

Codes have only a few subcode support weights.

Weight distributions are explicitly determined.

Abstract

Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight Griesmer $[\frac{q^k-1}{q-1},k ,q^{k-1}]_q$-linear codes and they meet all Griesmer bounds of the generalized Hamming weights of linear codes. All the subcodes with dimension $r$ of a $[\frac{q^k-1}{q-1},k ,q^{k-1}]_q$-simplex code have the same subcode support weight $\frac{q^{k-r}(q^r-1)}{q-1}$ for $1\leq r\leq k$. In this paper, we construct linear codes meeting the Griesmer bound of the $r$-generalized Hamming weight, such codes do not meet the Griesmer bound of the $j$-generalized Hamming weight for $1\leq j<r$. Moreover these codes have only few subcode support weights. The weight distribution and the subcode support weight distributions of these distance-optimal codes are determined. Linear codes constructed in this paper are natural generalizations of distance-optimal few-weight codes.