Griesmer and Optimal Linear Codes from the Affine Solomon-Stiffler Construction

TL;DR

This work establishes a geometric, affine Solomon-Stiffler construction that yields infinite families of Griesmer-bound-attaining and optimal linear codes over $\mathbb{F}_q$, including projective cases as special instances. By introducing affine and modified affine variants, the authors derive explicit parameters, weight distributions, and defect bounds, unifying and extending many known Griesmer and optimal codes (including several Grassl-listed cases) through a purely geometric framework. The paper also develops few-weight constructions, analyzes distance-optimality of subcodes and punctured codes of Griesmer codes, and demonstrates that many prior results in ring-based constructions can be recovered within this approach. Overall, the affine-Stiffler paradigm provides a versatile, scalable pathway to construct and analyze diverse families of distance-optimal and near-optimal codes with transparent geometric structure.

Abstract

In their fundamental paper published in 1965, G. Solomon and J. J. Stiffler invented infinite families of codes meeting the Griesmer bound. These codes are then called Solomon-Stiffler codes and have motivated various constructions of codes meeting or close the Griesmer bound. In this paper, we give a geometric construction of infinite families of affine and modified affine Solomon-Stiffler codes. Projective Solomon-Stiffler codes are special cases of our modified affine Solomon-Stiffler codes. Several infinite families of $q$-ary Griesmer, optimal, almost optimal two-weight, three-weight, four-weight and five-weight linear codes are constructed as special cases of our construction. Weight distributions of these Griesmer, optimal or almost optimal codes are determined. Many optimal linear codes documented in Grassl's list are re-constructed as (modified) affine Solomon-Stiffler codes. Several infinite families of optimal or Griesmer codes were constructed in two published papers in IEEE Transactions on Information Theory 2017 and 2019, via Gray images of codes over finite rings. Parameters and weight distributions of these Griesmer or optimal codes and very special case codes in our construction are the same. We also indicate that more general distance-optimal binary linear codes than that constructed in a recent paper of IEEE Transactions on Information Theory can be obtained directly from codimension one subcodes in binary Solomon-Stiffler codes.