The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes

Fengwei Li; Qin Yue; Xiaoming Sun

doi:10.3934/amc.2020049

Article Contents

2021, Volume 15, Issue 1: 131-153. Doi: 10.3934/amc.2020049

This issue Previous Article A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions Next Article Some properties of the cycle decomposition of WG-NLFSR

The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes

1.
School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, Shandong, 277160, China
2.
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China
3.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, China

Received: August 31, 2018

Revised: October 31, 2019

Early access: January 2020

Published: February 2021

The paper was supported by National Natural Science Foundation of China under Grants 11601475, 61772015, the foundation of Science and Technology on Information Assurance Labo- ratory under Grant KJ-17-010, china, and the foundation of innovative Science and technology for youth in universities of Shandong Province China under Grant 2019KJI001

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Abstract

Let $ l $ be a prime with $ l\equiv 3\pmod 4 $ and $ l\ne 3 $, $ N = l^m $ for $ m $ a positive integer, $ f = \phi(N)/2 $ the multiplicative order of a prime $ p $ modulo $ N $, and $ q = p^f $, where $ \phi(\cdot) $ is the Euler-function. Let $ \alpha $ be a primitive element of a finite field $ \Bbb F_{q} $, $ C_0^{(N,q)} = \langle \alpha^N\rangle $ a cyclic subgroup of the multiplicative group $ \Bbb F_q^* $, and $ C_i^{(N,q)} = \alpha^i\langle \alpha^N\rangle $ the cosets, $ i = 0,\ldots, N-1 $. In this paper, we use Gaussian sums to obtain the explicit values of $ \eta_i^{(N, q)} = \sum_{x \in C_i^{(N,q)}}\psi(x) $, $ i = 0,1,\cdots, N-1 $, where $ \psi $ is the canonical additive character of $ \Bbb F_{q} $. Moreover, we also compute the explicit values of $ \eta_i^{(2N, q)} $, $ i = 0,1,\cdots, 2N-1 $, if $ q $ is a power of an odd prime $ p $.

As an application, we investigate the weight distribution of a $ p $-ary linear code:

$ \mathcal{C}_{D} = \{C = ( \operatorname{Tr}_{q/p}(c x_1), \operatorname{Tr}_{q/p}(cx_2),\ldots, \operatorname{Tr}_{q/p}(cx_n)):c\in \Bbb{F}_{q}\}, $

where its defining set $ D $ is given by

$ D = \{x\in \Bbb{F}_{q}^{*}: \operatorname{Tr}_{q/p}(x^{\frac{q-1}{l^{m}}}) = 0\} $

and $ \operatorname{Tr}_{q/p} $ denotes the trace function from $ \Bbb F_{q} $ to $ \Bbb F_p $.

Keywords:

Mathematics Subject Classification: Primary: 11T23, 94B05; Secondary: 11L05.

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Table 1. Weight distribution of the code in Theorem 5.2

Weight	Frequency
0	1
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_0^{(l^{m-1},q)})$	$\frac{q-1}{l^{m-1}}$
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_i^{(l^{m-1},q)}),i = l^{m-1-k}u, u\in H_k^{(0)}$	$\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_{i'}^{(l^{m-1},q)}),i' = l^{m-1-k}u, u\in H_k^{(1)}$	$\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
	$k = 1,2,\ldots, m-1$

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Table 2. Weight distribution of the code in Theorem 5.3 $ (\frac{-1+\sqrt {-l}}2\equiv 0\pmod {\mathcal P_1})$

Weight	Frequency
$0$	$1$
$\frac{(p-1)(2l^mq-(l+1)(q-1))}{2pl^m}+\frac{p-1}p\eta_{0}^{(l^m, q)}+\frac{(l-1)(p-1)}{2p}\eta_{i'}^{(l^m,q)}$	$\frac{q-1}{l^{m}}$
$i'/l^{m-1}\in H_1^{(1)}$
$\frac{(p-1)(2l^mq-(l+1)(q-1))}{2pl^m}+\frac{p-1}p\eta_0^{(l^m, q)}+\frac{(l+1)(p-1)}{4p}\eta_{i}^{(l^m, q)}+\frac{(l-3)(p-1)}{4p}\eta_{i'}^{(l^m,q)}$	$\frac{q-1}{l^{m}}\cdot\frac{l-1}2$
$ i/l^{{m-1}}\in H_1^{(0)}, i'/l^{m-1}\in H_1^{(1)}$
$\frac{(p-1)(2l^mq-(l+1)(q-1))}{2pl^m}+\frac{(l+1)(p-1)}{4p}\eta_{i}^{(l^m, q)}+\frac{(l+1)(p-1)}{4p}\eta_{i'}^{(l^m,q)}$	$\frac{q-1}{l^{m}}\cdot\frac{l-1}2$
$i/l^{{m-1}}\in H_1^{(0)}, i'/l^{m-1}\in H_1^{(1)}$
$\frac{(p-1)(2l^mq-(l+1)(q-1))}{2pl^m}+\frac{(p-1)(l+1)}{2p}\eta_{i}^{(l^m, q)},i\in \cup_{k = 2}^m S_k$	$\frac{q-1}{l^{m}}\phi(l^k)$,
	$ k = 2,3,\ldots,m$,

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