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

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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  • 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 $.

    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
    $\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_0^{(l^{m-1},q)})$ $\frac{q-1}{l^{m-1}}$
    $\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_i^{(l^{m-1},q)}),i = l^{m-1-k}u, u\in H_k^{(0)}$ $\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
    $\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_{i'}^{(l^{m-1},q)}),i' = l^{m-1-k}u, u\in H_k^{(1)}$ $\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
    $k = 1,2,\ldots, m-1$
     | Show Table
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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$,
     | Show Table
    DownLoad: CSV
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