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

TL;DR

This paper introduces a geometric construction method for infinite families of affine Solomon-Stiffler codes, which include many optimal and Griesmer codes, and determines their weight distributions, unifying and extending previous results.

Contribution

It provides a new geometric construction of affine and modified affine Solomon-Stiffler codes, leading to many new and known optimal and Griesmer codes with explicit weight distributions.

Findings

Constructed infinite families of affine Solomon-Stiffler codes meeting the Griesmer bound.

Determined weight distributions for various optimal and almost optimal codes.

Reconstructed many codes from Grassl's list as (modified) affine Solomon-Stiffler codes.

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.