How many weights can a linear code have?

@article{Shi2019HowMW,
  title={How many weights can a linear code have?},
  author={Minjia Shi and Hongwei Zhu and Patrick Sol{\'e} and G{\'e}rard D. Cohen},
  journal={Designs, Codes and Cryptography},
  year={2019},
  volume={87},
  pages={87-95}
}
We study the combinatorial function L(k, q),  the maximum number of nonzero weights a linear code of dimension k over $${\mathbb {F}}_q$$Fq can have. We determine it completely for $$q=2,$$q=2, and for $$k=2,$$k=2, and provide upper and lower bounds in the general case when both k and q are $$\ge 3.$$≥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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