February  2021, 15(1): 131-153. doi: 10.3934/amc.2020049

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  September 2018 Revised  November 2019 Published  February 2021 Early access  January 2020

Fund Project: 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

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 $
.
Citation: Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
References:
[1]

L. Baumert and J. Mykkeltveit, Weight distributions of some irreducible cyclic codes, DSN Progr. Rep., 16 (1973), 128-131. 

[2]

B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, John Wiley & Sons Company, 1997.

[3]

H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02945-9.

[4]

C. S. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.

[5]

C. S. DingY. LiuC. L. Ma and L. W. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.

[6]

C. S. DingC. L. LiN. Li and Z. C. Zhou, Three-weight cyclic codes and their weight distributions, Discrete Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.

[7]

T. Feng and Q. Xiang, Strongly regular graphs from unions of cyclotomic classes, Journal of Combinatorial Theory Series B, 102 (2012), 982-995.  doi: 10.1016/j.jctb.2011.10.006.

[8]

Z. L. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491.  doi: 10.1109/LCOMM.2015.2455032.

[9]

Z. L. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002.

[10]

Z. L. Heng and Q. Yue, Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307-326.  doi: 10.1007/s10623-016-0222-7.

[11]

L. Q. HuQ. Yue and M. H. Wang, The linear complexity of Whiteman's generalize cyclotomic sequences of period $p^{m+1}q^{n+1}$, IEEE Trans. Inform. Theory, 58 (2012), 5534-5543.  doi: 10.1109/TIT.2012.2196254.

[12]

P. Langevin, Caluls de certaines sommes de Gauss, J. Number theory, 63 (1997), 59-64.  doi: 10.1006/jnth.1997.2078.

[13]

C. J. Li and Q. Yue, The Walsh transform of a class of monomial functions and cyclic codes, Cryptogr. Commun., 7 (2015), 217-228.  doi: 10.1007/s12095-014-0109-2.

[14]

C. J. LiQ. Yue and F. W. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[15]

F. W. LiQ. Yue and F. M. Liu, The weight distribution of a class of cyclic codes containing a subclass with optimal parameters, Finite Fields Appl., 45 (2017), 183-202.  doi: 10.1016/j.ffa.2016.12.004.

[16]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.

[17]

Y. W. Liu and Z. H. Liu, On some classes of codes with a few weights, Adv. Math. Commun., 12 (2018), 415-428.  doi: 10.3934/amc.2018025.

[18]

J. Q. Luo and K. Q. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.  doi: 10.1109/TIT.2008.2006424.

[19]

G. McGuire, On three weights in cyclic codes with two zeros, Finite Fields Appl., 10 (2004), 97-104.  doi: 10.1016/S1071-5797(03)00045-5.

[20]

G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.  doi: 10.4064/aa-39-3-251-264.

[21]

T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967.

[22]

Q. Y. WangK. L. DingD. D. Lin and R. Xue, A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315-322.  doi: 10.1007/s12095-015-0180-3.

[23]

Q. Y. WangK. L. Ding and R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100.  doi: 10.1109/LCOMM.2015.2431253.

[24]

X. Q. WangD. B. ZhengL. Hu and X. Y. Zeng, The weight distributions of two classes of binary cyclic codes, Finite Fields Appl., 34 (2015), 192-207.  doi: 10.1016/j.ffa.2015.01.012.

[25]

M. S. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.  doi: 10.1016/j.ffa.2012.06.001.

[26]

J. Yang and L. L. Xia, Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Math., 53 (2010), 2525-2542.  doi: 10.1007/s11425-010-3155-z.

[27]

J. YangM. S. XiongC. S. Ding and J. Q. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.

[28]

S. D. YangX. L. Kong and C. M. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[29]

S. D. Yang and Z.-A. Yao, Complete weight enumerators of a class of linear codes, Discrete Math., 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

[30]

Z. C. Zhou and C. S. Ding, A class of three-weight cyclic codes, Finite Fields Appli., 25 (2014), 79-93.  doi: 10.1016/j.ffa.2013.08.005.

[31]

Z. C. ZhouA. X. Zhang and C. S. Ding, The weight enumerator of three families of cyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 6002-6009.  doi: 10.1109/TIT.2013.2262095.

show all references

References:
[1]

L. Baumert and J. Mykkeltveit, Weight distributions of some irreducible cyclic codes, DSN Progr. Rep., 16 (1973), 128-131. 

[2]

B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, John Wiley & Sons Company, 1997.

[3]

H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02945-9.

[4]

C. S. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.

[5]

C. S. DingY. LiuC. L. Ma and L. W. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.

[6]

C. S. DingC. L. LiN. Li and Z. C. Zhou, Three-weight cyclic codes and their weight distributions, Discrete Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.

[7]

T. Feng and Q. Xiang, Strongly regular graphs from unions of cyclotomic classes, Journal of Combinatorial Theory Series B, 102 (2012), 982-995.  doi: 10.1016/j.jctb.2011.10.006.

[8]

Z. L. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491.  doi: 10.1109/LCOMM.2015.2455032.

[9]

Z. L. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002.

[10]

Z. L. Heng and Q. Yue, Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307-326.  doi: 10.1007/s10623-016-0222-7.

[11]

L. Q. HuQ. Yue and M. H. Wang, The linear complexity of Whiteman's generalize cyclotomic sequences of period $p^{m+1}q^{n+1}$, IEEE Trans. Inform. Theory, 58 (2012), 5534-5543.  doi: 10.1109/TIT.2012.2196254.

[12]

P. Langevin, Caluls de certaines sommes de Gauss, J. Number theory, 63 (1997), 59-64.  doi: 10.1006/jnth.1997.2078.

[13]

C. J. Li and Q. Yue, The Walsh transform of a class of monomial functions and cyclic codes, Cryptogr. Commun., 7 (2015), 217-228.  doi: 10.1007/s12095-014-0109-2.

[14]

C. J. LiQ. Yue and F. W. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[15]

F. W. LiQ. Yue and F. M. Liu, The weight distribution of a class of cyclic codes containing a subclass with optimal parameters, Finite Fields Appl., 45 (2017), 183-202.  doi: 10.1016/j.ffa.2016.12.004.

[16]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.

[17]

Y. W. Liu and Z. H. Liu, On some classes of codes with a few weights, Adv. Math. Commun., 12 (2018), 415-428.  doi: 10.3934/amc.2018025.

[18]

J. Q. Luo and K. Q. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.  doi: 10.1109/TIT.2008.2006424.

[19]

G. McGuire, On three weights in cyclic codes with two zeros, Finite Fields Appl., 10 (2004), 97-104.  doi: 10.1016/S1071-5797(03)00045-5.

[20]

G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.  doi: 10.4064/aa-39-3-251-264.

[21]

T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967.

[22]

Q. Y. WangK. L. DingD. D. Lin and R. Xue, A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315-322.  doi: 10.1007/s12095-015-0180-3.

[23]

Q. Y. WangK. L. Ding and R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100.  doi: 10.1109/LCOMM.2015.2431253.

[24]

X. Q. WangD. B. ZhengL. Hu and X. Y. Zeng, The weight distributions of two classes of binary cyclic codes, Finite Fields Appl., 34 (2015), 192-207.  doi: 10.1016/j.ffa.2015.01.012.

[25]

M. S. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.  doi: 10.1016/j.ffa.2012.06.001.

[26]

J. Yang and L. L. Xia, Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Math., 53 (2010), 2525-2542.  doi: 10.1007/s11425-010-3155-z.

[27]

J. YangM. S. XiongC. S. Ding and J. Q. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.

[28]

S. D. YangX. L. Kong and C. M. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[29]

S. D. Yang and Z.-A. Yao, Complete weight enumerators of a class of linear codes, Discrete Math., 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

[30]

Z. C. Zhou and C. S. Ding, A class of three-weight cyclic codes, Finite Fields Appli., 25 (2014), 79-93.  doi: 10.1016/j.ffa.2013.08.005.

[31]

Z. C. ZhouA. X. Zhang and C. S. Ding, The weight enumerator of three families of cyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 6002-6009.  doi: 10.1109/TIT.2013.2262095.

Table 1.  Weight distribution of the code in Theorem 5.2
Weight Frequency
0 1
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_0^{(l^{m-1},q)})$ $\frac{q-1}{l^{m-1}}$
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_i^{(l^{m-1},q)}),i = l^{m-1-k}u, u\in H_k^{(0)}$ $\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_{i'}^{(l^{m-1},q)}),i' = l^{m-1-k}u, u\in H_k^{(1)}$ $\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
$k = 1,2,\ldots, m-1$
Weight Frequency
0 1
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_0^{(l^{m-1},q)})$ $\frac{q-1}{l^{m-1}}$
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_i^{(l^{m-1},q)}),i = l^{m-1-k}u, u\in H_k^{(0)}$ $\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
$\frac{p-1}p(q-\frac{q-1}{l^{m-1}}+\eta_{i'}^{(l^{m-1},q)}),i' = l^{m-1-k}u, u\in H_k^{(1)}$ $\frac{q-1}{l^{m-1}}\cdot\frac{\phi(l^k)}2$
$k = 1,2,\ldots, m-1$
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$,
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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