Griesmer codes are linear codes meeting the Griesmer bound. A linear code is called Δ-divisible if the weights of all codewords are divisible by Δ. In this paper, we investigate the divisibility of Griesmer codes. Let C be an n , k , d q Griesmer code with q = p f , where p is a prime and f ≥ 1 is an integer. In earlier work, Ward proved: (1) for prime fields (i.e., q = p ), if p e | d , then C is p e -divisible; (2) if q | d , then C is p -divisible; (3) when q = 4 , if 2 e | d , then C is ⌈ 2 e − 1 ⌉ -divisible. Based on these results, Ward conjectured that if p e | d , then C is ⌈ p e − ( f − 1 ) ⌉ -divisible. In the present work, we obtain two new results on divisibility of a Griesmer n , k , d q code C : (a) if q e | d , then C is p e -divisible; (b) if p e | d , then C is ⌈ p e − ( f − 1 ) ( q − 2 ) ⌉ -divisible. To prove these results, we first show that any n , k , d q Griesmer code C admits an ordered basis consisting of k codewords such that the first i of them span a Griesmer subcode for any 1 ≤ i ≤ k , and any k − 1 of them span a Griesmer subcode. This special basis is central to our proofs. Secondly, we derive some inequalities of the p -adic valuations involving binomial coefficients. By applying Ward's divisibility criterion to the aforementioned basis, we extend Result (1) and (2) to Result (a). Finally, using the geometric approach and the properties of the special basis, we reduce the divisibility analysis of C to the case ν p ( d ) < f ( q − 2 ) . Combining this reduction with Result (a), we establish Result (b).
Deng et al. (Fri,) studied this question.