# 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.

## 12 Citations

A note on full weight spectrum codes

- Mathematics, Computer ScienceArXiv
- 2018

In this brief communication, necessary and sufficient conditions for the existence of linear full weight spectrum (FWS) codes are determined.

1 2 A pr 2 01 8 On linear codes and distinct weights

- 2018

We provide a construction for linear codes attaining the maximum possible number of distinct weights. We then introduce the related problem of determining the existence of linear codes with an…

1 7 A pr 2 01 8 Maximum weight spectrum codes

- 2018

In the recent work [7], a combinatorial problem concerning linear codes over a finite field Fq was introduced. In that work the authors studied the weight set of an [n, k]q linear code, that is the…

How Many Weights Can a Cyclic Code Have?

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2020

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper…

How Many Weights Can a Quasi-Cyclic Code Have?

- Mathematics, Computer ScienceIEEE Transactions on Information Theory
- 2020

The largest number of nonzero weights of quasi-cyclic codes is investigated, and the smallest index for which a <inline-formula> <tex-math notation="LaTeX">$q$ </tex-Math></inline- formula>-ary Reed-Muller code is quasi- cyclic is determined.

Maximum Weight Spectrum Codes

- Computer Science, MathematicsAdv. Math. Commun.
- 2019

This work establishes the truth of the conjecture that the bound is sharp for every prime power q and every positive integer k, and establishes some lower bounds on the length of codes that satisfy the conjecture.

ON THE GENERALISATION OF SIDEL’NIKOV’S THEOREM TO $q$ -ARY LINEAR CODES

- MathematicsBulletin of the Australian Mathematical Society
- 2019

We generalise Sidel’nikov’s theorem from binary codes to $q$ -ary codes for $q>2$ . Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by…

Maximum weight spectrum codes with reduced length

- Mathematics, Computer ScienceArXiv
- 2018

By an averaging argument, this work shows the existence of MWS codes of even shorter length, constructed from quasi-minimal codes, thus obtaining of much shorter length than hitherto known.

On linear codes and distinct weights

- Mathematics, Computer ScienceArXiv
- 2018

The related problem of determining the existence of linear codes with an arbitrary number of distinct non-zero weights is introduced, and a solution is completely determine in the binary case.

Weight distribution of a subclass of Z2-double cyclic codes

- Computer Science, MathematicsFinite Fields Their Appl.
- 2019

The weight distribution of Z 2 -double cyclic codes of length m 0 + m 1 for a special case are determined explicitly.

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