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}
}
• M. Shi, +1 author G. Cohen
• Published 1 February 2018
• Mathematics, Computer Science
• Designs, Codes and Cryptography
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.
12 Citations
A note on full weight spectrum codes
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1 7 A pr 2 01 8 Maximum weight spectrum codes
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Maximum Weight Spectrum Codes
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ON THE GENERALISATION OF SIDEL’NIKOV’S THEOREM TO \$q\$ -ARY LINEAR CODES
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Maximum weight spectrum codes with reduced length
• Mathematics, Computer Science
ArXiv
• 2018
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On linear codes and distinct weights
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Weight distribution of a subclass of Z2-double cyclic codes
• Computer Science, Mathematics
Finite Fields Their Appl.
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