Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over $\mathbb{F}_{q^\ell}$ of a $q$-ary linear code with significant combinatorial properties, which can determine the support weight distribution of this linear code. The Solomon-Stiffler codes are a family of famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In this paper, we determine the weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes using some combinatorial properties of subspaces. As a result, we determine the support weight distributions of the projective Solomon-Stiffler codes. In particular, we determine the weight hierarchies of the projective Solomon-Stiffler codes.

翻译：确定码的权分布是编码理论中一个古老且基础的研究课题，已被深入探讨。1977年，Helleseth、Kløve和Mykkeltveit提出了q元线性码在$\mathbb{F}_{q^\ell}$上的升码的权枚举多项式，该多项式具有显著的组合性质，可用于确定该线性码的支持权分布。Solomon-Stiffler码是一类著名的Griesmer码，由Solomon和Stiffler于1965年提出。本文利用子空间的某些组合性质，确定了射影Solomon-Stiffler码的升码的权枚举多项式。由此，我们进一步确定了射影Solomon-Stiffler码的支持权分布。特别地，我们计算了射影Solomon-Stiffler码的权层次。