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How many weights can a linear code have?

Abstract

We study the combinatorial function L(kq),  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(nkq),  as well as nonlinear analogues N(Mq) and N(nMq),  are also introduced and studied.

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Notes

  1. After submission of this article, a proof was found in [1].

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Authors and Affiliations

Authors

Corresponding author

Correspondence to Minjia Shi.

Additional information

Communicated by J. Jedwab.

This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

Appendix: numerical examples

Appendix: numerical examples

We provide lower bounds on L(kq) 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 \(n=6,000,000\)

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Shi, M., Zhu, H., Solé, P. et al. How many weights can a linear code have?. Des. Codes Cryptogr. 87, 87–95 (2019). https://doi.org/10.1007/s10623-018-0488-z

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  • DOI: https://doi.org/10.1007/s10623-018-0488-z

Keywords

  • Linear codes
  • Hamming weight
  • Perfect difference sets

Mathematics Subject Classification

  • 94B05
  • 05B10