Griesmer and Optimal Linear Codes From the Affine Solomon–Stiffler Construction

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. However weight distributions of Solomon-Stiffler codes have been only determined for very special cases. In this paper, we give a geometric construction 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>-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 explicitly. 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 Shi et, al., IEEE Trans. Inf. Theory, vol. 63, no. 10, 2017, and in Liu et, al., IEEE Trans. Inf. Theory, vol. 65, no. 5, 2019, via Gray images of codes over finite rings. Parameters and weight distributions of these Griesmer or optimal codes can be realized as very special cases in our construction. We also indicate that more general optimal binary linear codes than that constructed in Mondal, IEEE Trans. Inf. Theory, vol. 70, no. 7, 2024, can be obtained from subcodes of codimension one in the binary Solomon-Stiffler codes.