A Bound for the Maximum Weight of a Linear Code Simeon Ball and Aart Blokhuis

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 ≤ (n− d)p− e(p− 1), where e ∈ {0, 1, . . . , k− 2} is maximal with the property that ( n− d e ) 6≡ 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.