A Bound for the Maximum Weight of a Linear Code

Abstract

It is shown that the parameters of a linear code over ${\mathbb F}_q$ 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 $m \leq (n-d)p-e(p-1)$, where $e \in \{0,1,\ldots,k-2 \}$ is maximal with the property that ${n-d \choose e} \not\equiv 0 \pmod{p^{k-1-e}}.$ Thus, if $C$ contains a codeword of weight $n$, then $n \geq d/(p-1)+d+e$. 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.