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.

翻译：1965年，G. Solomon与J. J. Stiffler在其奠基性论文中提出了满足格里瑟默界的无限码族，这些码后被称作所罗门-斯蒂夫勒码，并推动了满足或接近格里瑟默界的各类码构造研究。本文从几何角度构建了仿射及修正仿射所罗门-斯蒂夫勒码的无限族，其中射影所罗门-斯蒂夫勒码可视为修正仿射所罗门-斯蒂夫勒码的特例。通过本构造的特殊情形，我们构建了多类无限族的$q$元格里瑟默码、最优码、近似最优二重码、三重码、四重码与五重码线性码，并确定了这些格里瑟默码、最优码或近似最优码的重量分布。Grassl列表中的许多最优线性码均可重构为（修正）仿射所罗门-斯蒂夫勒码。IEEE Transactions on Information Theory 2017与2019年刊载的两篇论文中，通过有限环上码的格雷映射构建了若干最优或格里瑟默码的无限族，这些格里瑟默码或最优码的参数、重量分布与本构造中极端特例码完全一致。我们进一步指出，相较于IEEE Transactions on Information Theory近期论文所构建的码，更一般的距离最优二元线性码可直接从二元所罗门-斯蒂夫勒码的余维一子码中获得。