Two-weight and three-weight linear codes constructed from Weil sums

Tonghui Zhang; Hong Lu; Shudi Yang

doi:10.3934/mfc.2021041

Article Contents

2022, Volume 5, Issue 2: 129-144. Doi: 10.3934/mfc.2021041

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Two-weight and three-weight linear codes constructed from Weil sums

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

^* Corresponding author: Shudi Yang
^* Corresponding author: Shudi Yang

Received: October 2021

Revised: December 2021

Early access: January 2022

Published: May 2022

The work is partially supported by National Natural Science Foundation of China under Grant 12071247 and by Research and Innovation Fund for Graduate Dissertations of Qufu Normal University in 2021 under Grant LWCXS202133

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Abstract

Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 11T71.

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Table 1. The weight distribution of $ C_{D_1} $

weight	multiplicity
0	1
$ q^{2m-2}\left(q-1 \right) -q^{3s-2} $	$ \left(q^m-q^{m-1}+q^s-q^{s-1}\right) \left(q-1\right) $
$ q^{2m-2}\left( q-1\right) $	$ q^m\left(q^m-q+1 \right)-1 $
$ \left( q^{2m-2}+q^{3s-2}\right)\left(q-1 \right) $	$ \left(q^{m-1}-q^s+q^{s-1}\right) \left(q-1\right) $

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Table 2. The weight distribution of $ C_{D_2} $ when $ q\equiv 1\pmod{4} $

weight	multiplicity
0	1
$ \; \qquad q^{2m-2}\left( q-1\right) $	$ \left( q^m-1\right)\left( q^{m-1}+1\right) \qquad \; $
$ \; \qquad q^{m-1}\left( q^{m-1}+1\right) \left(q-1 \right) $	$ q^{m-1}\left( q^m-1\right) \left( q-1\right) \qquad \; $

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Table 3. The weight distribution of $ C_{D_2} $ when $ q\equiv 3\pmod{4} $

weight	multiplicity
0	1
$ \;q^{m-1}\left( q^{m-1}+(-1)^s\right) \left(q-1 \right) $	$ q^{m-1}\left( q^m-(-1)^s\right) \left( q-1\right) \; $
$ \;q^{2m-2}\left( q-1\right) $	$ \left( q^m-(-1)^s\right)\left( q^{m-1}+(-1)^s\right) \; $

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Table 4. The weight distribution of $ C_{D_2} $ when $ q $ is even and $ q>2 $

weight	multiplicity
0	1
$ \left( q^{2m-2}-q^{3s-2}\right)\left(q-1 \right) $	$ \dfrac{1}{2}\left(q^m+q^s \right) \left(q-1\right) $
$ q^{2m-2}\left( q-1\right) $	$ q^m\left(q^m-q+1 \right)-1 $
$ \left( q^{2m-2}+q^{3s-2}\right) \left(q-1 \right) $	$ \dfrac{1}{2}\left(q^m-q^s \right) \left(q-1\right) $

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