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

Fengwei Li, ,School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, Shandong, 277160, China, Qin Yue, Xiaoming Sun, ,State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China, ,Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, China
2019 Advances in Mathematics of Communications  

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Let l be a prime with l ≡ 3 (mod 4) and l ̸ = 3, N = l m for m a positive integer, f = ϕ(N )/2 the multiplicative order of a prime p modulo N , and q = p f , where ϕ(·) is the Euler-function. Let α be a primitive element of a finite field Fq, C (N,q) 0 = ⟨α N ⟩ a cyclic subgroup of the multiplicative group F * q , and C (N,q) i = α i ⟨α N ⟩ the cosets, i = 0, . . . , N − 1. In this paper, we use Gaussian sums to obtain the explicit values of η (N,q) i = ∑ x∈C (N,q) i ψ(x), i = 0, 1, · · · , N
more » ... , where ψ is the canonical additive character of Fq. Moreover, we also compute the explicit values of η (2N,q) i , i = 0, 1, · · · , 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: C D = {C = (Tr q/p (cx 1 ), Tr q/p (cx 2 ), . . . , Tr q/p (cxn)) : c ∈ Fq}, where its defining set D is given by l m ) = 0} and Tr q/p denotes the trace function from Fq to Fp.
doi:10.3934/amc.2020049 fatcat:h3vmygtedzgoll3vddihhjpwaq
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Fengwei Li, ,School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, Shandong, 277160, China, Qin Yue, Xiaoming Sun, ,State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China, ,Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, China. "The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes." Advances in Mathematics of Communications 15.1 (2019) 0-0