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  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.   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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

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

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

[2]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[3]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79

[4]

Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021016

[5]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017

[6]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[7]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[8]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[9]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

[10]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[11]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[12]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[13]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[14]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[15]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006

[16]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617

[17]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (116)
  • HTML views (537)
  • Cited by (0)

Other articles
by authors

[Back to Top]