How many weights can a linear code have ?

25 Apr 2018 · Shi Minjia, Zhu Hongwei, Solé Patrick, Cohen Gérard D. ·

We study the combinatorial function $L(k,q),$ the maximum number of nonzero weights a linear code of dimension $k$ over $\F_q$ can have. We determine it completely for $q=2,$ and for $k=2,$ and provide upper and lower bounds in the general case when both $k$ and $q$ are $\ge 3.$ A refinement $L(n,k,q),$ as well as nonlinear analogues $N(M,q)$ and $N(n,M,q),$ are also introduced and studied.

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How many weights can a linear code have ?

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