A bound for the maximum weight of a linear code

S.M. Ball, A. Blokhuis

Research output: Contribution to journalArticleAcademicpeer-review

7 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)575-583
JournalSIAM Journal on Discrete Mathematics
Volume27
Issue number1
DOIs
Publication statusPublished - 2013

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