How many weights can a linear code have?

Abstract

We study the combinatorial function L(k, q),  the maximum number of nonzero weights a linear code of dimension k over ${\mathbb {F}}_q$ 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.

Appendix: numerical examples

We provide lower bounds on L(k, q) by computing the number of weights in long random codes produced by the computer package Magma [8]. We give some numerical examples in Table 1 about the lower bound of Proposition 4.

Table 1 Proposition 4

When n is in the millions, we can find linear $[n,k]_q$-codes that meet the upper bound in Proposition 2: see Table 2.

Table 2 $n=6,000,000$