Optimal Few-GHW Linear Codes and Their Subcode Support Weight Distributions
Xu Pan, Hao Chen, Hongwei Liu, Shengwei Liu
TL;DR
The paper addresses the problem of constructing distance-optimal, few-weight linear codes with strong generalized Hamming weight properties. It develops four infinite families of distance-optimal $r$-Griesmer codes via modified Solomon-Stiffler and simplex-complement constructions, and provides explicit formulas for their $r$-generalized Hamming weights and $r$-subcode support weight distributions. The main contributions include the complete determination of weight and subcode-weight distributions in many cases, along with proofs of distance-optimality and Griesmer-code status, supported by concrete examples and MAGMA computations. These results deepen the understanding of weight hierarchies and subcode structures in optimal few-weight codes, offering systematic methods for generating such codes and suggesting directions for further exploration in code design.
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.