Complete weight enumerators of a class of linear codes over finite fields

Shudi Yang; Xiangli Kong; Xueying Shi

doi:10.3934/amc.2020045

Article Contents

2021, Volume 15, Issue 1: 99-112. Doi: 10.3934/amc.2020045 open access

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Complete weight enumerators of a class of linear codes over finite fields

1.
School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

^*Corresponding author: Xiangli Kong
^*Corresponding author: Xiangli Kong

Received: March 31, 2019

Early access: November 2019

Published: February 2021

The work is partially supported by the National Natural Science Foundation of China (11701317, 11801303, 11571380) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04). This work is also partially supported by Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (17KJB110018)

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Abstract

We investigate a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. These codes have at most three weights and some of them are almost optimal so that they are suitable for applications in secret sharing schemes. This is a supplement of the results raised by Wang et al. (2017) and Kong et al. (2019).

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

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Table 1. Weight distribution of $ \mathcal{C} _{D_0} $ for odd $ e $

Weight $ w $	Frequency $ A_w $
$ 0 $	$ 1 $
$ (p-1) p^{e-2} $	$ p^{e-1} -1 $
$ (p-1) (p^{e-2}- p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $	$ \dfrac{p-1}{2p} {\big( {{ q+ \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
$ (p-1) (p^{e-2}+ p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $	$ \dfrac{p-1}{2p} {\big( {{ q- \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $

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Table 2. Weight distribution of $ \mathcal{C} _{D_0} $ for even $ e $

Weight $ w $	Frequency $ A_w $
$ 0 $	$ 1 $
$ (p-1) p^{e-2} $	$ p^{e-1} + (p-1)p^{-1} \varepsilon_0 \eta_e(a)-1 $
$ (p-1) (p^{e-2}+ p^{-1} \varepsilon_0 \eta_e(a) ) $	$ (p-1) {\big( {{p^{e-1}- p^{-1} \varepsilon_0 \eta_e(a)}} \big) } $

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Table 3. Weight distribution of $ \mathcal{C} _{D_c} $ for odd $ e $ and $ c \neq 0 $

Weight $ w $	Frequency $ A_w $
$ 0 $	$ 1 $
$ (p-1) p^{e-2} $	$ p^{e-1} -1 $
$ n_c-p^{e-2} + p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $	$ \frac{p-1}{2} n_c $
$ n_c-p^{e-2} - p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $	$ \frac{p-1}{2}{\big( {{p^{e-1} - p^{-1}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) }} \big) } $

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Table 4. Weight distribution of $ \mathcal{C} _{D_c} $ for even $ e $ and $ c\neq 0 $

Weight $ w $	Frequency $ A_w $
$ 0 $	$ 1 $
$ (p-1) p^{e-2} $	$ \frac{p+1}{2}p^{e-1}+\frac{p-1}{2} p^{-1}\varepsilon_0 \eta_e(a)-1 $
$ (p-1)p^{e-2} -2 p^{-1}\varepsilon_0 \eta_e(a) $	$ \frac{p-1}{2}n_c $

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