American Institute of Mathematical Sciences

Advanced
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

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

Download as excel

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$,
Table Options

Download as excel

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$,
[1]

René Henrion. Gradient estimates for Gaussian distribution functions: application to probabilistically constrained optimization problems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 655-668. doi: 10.3934/naco.2012.2.655
[2]

Lin Yi, Xiangyong Zeng, Zhimin Sun, Shasha Zhang. On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021019
[3]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[4]

Zhenghong Qiu, Jianhui Huang, Tinghan Xie. Linear-Quadratic-Gaussian mean-field controls of social optima. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021047
[5]

Pierre Gervais. A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights. Kinetic & Related Models, 2021, 14 (4) : 725-747. doi: 10.3934/krm.2021022
[6]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025
[7]

Marie Turčičová, Jan Mandel, Kryštof Eben. Score matching filters for Gaussian Markov random fields with a linear model of the precision matrix. Foundations of Data Science, 2021, 3 (4) : 789-820. doi: 10.3934/fods.2021030
[8]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[9]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure & Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411
[10]

Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 55-79. doi: 10.3934/nhm.2007.2.55
[11]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[12]

Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. On the weight distribution of the cosets of MDS codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021042
[13]

Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems & Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397
[14]

Andreas Asheim, Alfredo Deaño, Daan Huybrechs, Haiyong Wang. A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 883-901. doi: 10.3934/dcds.2014.34.883
[15]

Wenxiong Chen, Congming Li. Some new approaches in prescribing gaussian and salar curvature. Conference Publications, 1998, 1998 (Special) : 148-159. doi: 10.3934/proc.1998.1998.148
[16]

Luca Di Persio, Giacomo Ziglio. Gaussian estimates on networks with applications to optimal control. Networks & Heterogeneous Media, 2011, 6 (2) : 279-296. doi: 10.3934/nhm.2011.6.279
[17]

Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
[18]

Sheng Zhang, Xiu Yang, Samy Tindel, Guang Lin. Augmented Gaussian random field: Theory and computation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021098
[19]

Johannes Hertrich, Dang-Phuong-Lan Nguyen, Jean-Francois Aujol, Dominique Bernard, Yannick Berthoumieu, Abdellatif Saadaldin, Gabriele Steidl. PCA reduced Gaussian mixture models with applications in superresolution. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021053
[20]

Víctor Almeida, Jorge J. Betancor. Variation and oscillation for harmonic operators in the inverse Gaussian setting. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021183

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (277)
  • HTML views (543)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy