A construction of $ p $-ary linear codes with two or three weights

Hongming Ru; Chunming Tang; Yanfeng Qi; Yuxiao Deng

doi:10.3934/amc.2020039

Article Contents

2021, Volume 15, Issue 1: 9-22. Doi: 10.3934/amc.2020039

This issue Previous Article New bounds on the minimum distance of cyclic codes Next Article Golay complementary sets with large zero odd-periodic correlation zones

A construction of $ p $-ary linear codes with two or three weights

1.
School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China
2.
School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China
3.
Electronics Technology Institute, Beijing, 100195, China

^* Corresponding author: Chunming Tang
^* Corresponding author: Chunming Tang

Received: January 2019

Revised: July 2019

Early access: November 2019

Published: February 2021

This work is supported by the National Natural Science Foundation of China (Grant No. 11871058, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005)

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Abstract

Applied in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs, linear codes attract much interest. We consider the construction of linear codes with two or three weights. Let $ m_1,\ldots, m_t $ be $ t $ positive integers and $ T = \mathbb{F}_{q_1}\times \cdots \times \mathbb{F}_{q_t} $, where $ q_i = p^{m_i} $ for $ 1\leq i\leq t $ and $ p $ is an odd prime. A linear code

$ \begin{equation*} \mathcal{C}_D = \{ \mathbf{c}(\mathbf{a}): \mathbf{a} = (a_1,\ldots,a_t)\in T\}, \end{equation*} $

can be constructed by a defining set $ D $, where $ D $ is a subset of $ T $ and $ \mathbf{c}(\mathbf{a}) = (\sum_{i = 1}^{t}\mathrm{Tr}_1^{m_i}(a_ix_i))_{ \mathbf{x} = (x_1,\ldots,x_t)\in D} $. We construct linear codes with two or three weights from the following three defining sets:

● $ D_0 = \{\mathbf{x}\in T\backslash \{\mathbf{0}\}: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2) = 0\} $,

● $ D_{SQ} = \{\mathbf{x}\in T: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2)\in SQ\} $,

● $ D_{NSQ} = \{\mathbf{x}\in T: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2)\in NSQ\} $,

where $ SQ $ is the set of all the squares in $ \mathbb{F}_p^* $ and $ NSQ $ is the set of all the nonsquares in $ \mathbb{F}_p^* $. We also determine the weight distributions of these codes. The punctured codes of codes from the defining set $ D_0 $ contain optimal codes meeting certain bounds. This paper generalizes results of [22].

Keywords:

Mathematics Subject Classification: Primary: 94B05, 11T71, 11T23.

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Table 1. The weight distribution of $ \mathcal{C}_{D_0} $ for $ \sum_{i = 1}^tm_i $ odd

Weight	Frequency
0	1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $	$ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)(p^{\sum_{i=1}^tm_i-2}- p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)(p^{\sum_{i=1}^tm_i-2}+ p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $

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Table 2. The weight distribution of $ \mathcal{C}_{D_0} $ for $ \sum_{i = 1}^tm_i $ even

Weight	Frequency
0	1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $	$ p^{\sum_{i=1}^tm_i-1} +M (p-1)p^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ (p-1)(p^{\sum_{i=1}^tm_i-2} +Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $	$ (p-1)(p^{\sum_{i=1}^tm_i-1} -Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* M=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}$

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Table 3. The weight distribution of $ \mathcal{C}_{\overline{D}_0} $ for $ \sum_{i = 1}^tm_i $ odd

Weight	Frequency
0	1
$ p^{\sum_{i=1}^tm_i-2} $	$ p^{\sum_{i=1}^t m_i-1}-1 $
$ p^{\sum_{i=1}^tm_i-2}- p^{\frac{\sum_{i=1}^tm_i-3}{2}} $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ p^{\sum_{i=1}^tm_i-2}+ p^{\frac{\sum_{i=1}^tm_i-3}{2}} $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $

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Table 4. The weight distribution of $ \mathcal{C}_{\overline{D}_0} $ for $ \sum_{i = 1}^tm_i $ even

Weight	Frequency
0	1
$ p^{\sum_{i=1}^tm_i-2} $	$ p^{\sum_{i=1}^tm_i-1} +M (p-1)p^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ p^{\sum_{i=1}^tm_i-2} +Mp^{\frac{\sum_{i=1}^tm_i-2}{2}} $	$ (p-1)(p^{\sum_{i=1}^tm_i-1} -Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* M=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}$

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Table 5. The weight distribution of $ \mathcal{C}_{D_{SQ}} $ for $ \sum_{i = 1}^tm_i $ odd

Weight	Frequency
0	1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $	$ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} +A (p^{\frac{\sum_{i=1}^tm_i-1}{2}} + p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)p^{\sum_{i=1}^tm_i-2} +A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} - p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$^* A=(-1) \frac{(p-1)\left(\sum_{i=1}^{t} m_{i}+3\right)}{4}+t+1.$

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Table 6. The weight distribution of $ \mathcal{C}_{D_{SQ}} $ for $ \sum_{i = 1}^tm_i $ even

Weight	Frequency
0	1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $	$ \frac{p+1}{2}p^{\sum_{i=1}^tm_i-1} +\frac{p-1}{2}Bp^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -2Bp^{\frac{\sum_{i=1}^tm_i-2}{2}} $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} -Bp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* B=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}.$

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Table 7. The weight distribution of $ \mathcal{C}_{D_{NSQ}} $ for $ \sum_{i = 1}^tm_i $ odd

Weight	Frequency
0	1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $	$ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} - p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} + p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $	$ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$^* A=(-1) \frac{(p-1)\left(\sum_{i=1}^{t} m_{i}+3\right)}{4}+t+1.$

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