If q^e divides the minimum distance of a Griesmer code over F_q, all codeword weights are divisible by p^e; a separate geometric argument gives a weaker but new divisibility bound when only p^e divides the minimum distance.
Bonisoli,Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin.18(1984), 181–186
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Divisibility of Griesmer Codes
If q^e divides the minimum distance of a Griesmer code over F_q, all codeword weights are divisible by p^e; a separate geometric argument gives a weaker but new divisibility bound when only p^e divides the minimum distance.