It is shown that the parameters of a linear code over Fq of length n, dimension k, minimum weight d, and maximum weight m satisfy a certain congruence relation. In the case that q = p is a prime, this leads to the bound m &le (n-d)p-e(p-1), where e {0, 1,.., k-2} is maximal with the property that (n-de) 0 (mod pk-1-e). Thus, if C contains a codeword of weight n, then n-d/(p-1)+d+e. The results obtained for linear codes are translated into corresponding results for (n, t)-arcs and t-fold blocking sets of AG(k-1, q). The bounds obtained in these spaces are better than the known bounds for these geometrical objects for many parameters

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