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

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doi: 10.3934/amc.2020049

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

Fengwei Li ^1,2, , Qin Yue ^3, and Xiaoming Sun ^1,

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 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} $

$ \Bbb F_p $

Keywords: Gaussian period, Gaussian sum, cyclotomic polynomial, weight distribution, Linear code.

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

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, 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. Google Scholar
[2]	B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, John Wiley & Sons Company, 1997. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	C. S. Ding, Y. Liu, C. 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. Google Scholar
[6]	C. S. Ding, C. L. Li, N. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	L. Q. Hu, Q. 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. Google Scholar
[12]	P. Langevin, Caluls de certaines sommes de Gauss, J. Number theory, 63 (1997), 59-64. doi: 10.1006/jnth.1997.2078. Google Scholar
[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. Google Scholar
[14]	C. J. Li, Q. 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. Google Scholar
[15]	F. W. Li, Q. 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. Google Scholar
[16]	R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967. Google Scholar
[22]	Q. Y. Wang, K. L. Ding, D. 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. Google Scholar
[23]	Q. Y. Wang, K. L. Ding and R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100. doi: 10.1109/LCOMM.2015.2431253. Google Scholar
[24]	X. Q. Wang, D. B. Zheng, L. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	J. Yang, M. S. Xiong, C. 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. Google Scholar
[28]	S. D. Yang, X. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	Z. C. Zhou, A. 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. Google Scholar

show all references

References:

[1]	L. Baumert and J. Mykkeltveit, Weight distributions of some irreducible cyclic codes, DSN Progr. Rep., 16 (1973), 128-131. Google Scholar
[2]	B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, John Wiley & Sons Company, 1997. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	C. S. Ding, Y. Liu, C. 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. Google Scholar
[6]	C. S. Ding, C. L. Li, N. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	L. Q. Hu, Q. 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. Google Scholar
[12]	P. Langevin, Caluls de certaines sommes de Gauss, J. Number theory, 63 (1997), 59-64. doi: 10.1006/jnth.1997.2078. Google Scholar
[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. Google Scholar
[14]	C. J. Li, Q. 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. Google Scholar
[15]	F. W. Li, Q. 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. Google Scholar
[16]	R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967. Google Scholar
[22]	Q. Y. Wang, K. L. Ding, D. 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. Google Scholar
[23]	Q. Y. Wang, K. L. Ding and R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100. doi: 10.1109/LCOMM.2015.2431253. Google Scholar
[24]	X. Q. Wang, D. B. Zheng, L. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	J. Yang, M. S. Xiong, C. 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. Google Scholar
[28]	S. D. Yang, X. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	Z. C. Zhou, A. 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. Google Scholar

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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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$,

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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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