# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022032
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm

 1 Department of Physical, Mathematical and Engineering Sciences, University of Chester, Exton Park, Chester CH1 4AR, England 2 Tarsus University, Faculty of Engineering, Department of Natural and Mathematical Sciences, Mersin, Turkey 3 Tarsus University, Faculty of Engineering, Department of Computer Engineering, Mersin, Turkey

* Corresponding author

Received  November 2021 Revised  March 2022 Early access April 2022

In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $k^{th}$-range neighbours, and search for binary $[72, 36, 12]$ self-dual codes. In particular, we present six generator matrices of the form $[I_{36} \ | \ \tau_6(v)],$ where $I_{36}$ is the $36 \times 36$ identity matrix, $v$ is an element in the group matrix ring $M_6(\mathbb{F}_2)G$ and $G$ is a finite group of order 6, to which we employ the proposed algorithm and search for binary $[72, 36, 12]$ self-dual codes directly over the finite field $\mathbb{F}_2$. We construct 1471 new Type I binary $[72, 36, 12]$ self-dual codes with the rare parameters $\gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32$ in their weight enumerators.

Citation: Adrian Korban, Serap Sahinkaya, Deniz Ustun. New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm. Advances in Mathematics of Communications, doi: 10.3934/amc.2022032
##### References:
 [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125. [2] J. R. Cuevas, H. J. Wang, Y. C. Lai and Y. C. Liang, Virus optimization algorithm: A novel metaheuristic for solving continuous optimization problems, The 10th Asia Pacific Industrial Engineering Management System Conference, (2009), 2166–2174. [3] S. T. Dougherty, The neighbor graph of binary self-dual codes, Des. Codes and Cryptogr., 90 (2022), 409-425.  doi: 10.1007/s10623-021-00985-2. [4] S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over frobenius rings, Cryptogr. Commun., 12, (2020), 127–146. doi: 10.1007/s12095-019-00380-8. [5] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Adv. Math. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037. [6] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, New extremal self-dual binary codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbors, Int. J. Inf. Coding Theory, 5 (2018), 211-226. [7] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Composite matrices from group rings, composite $G$-codes and constructions of self-dual codes, Des. Codes Cryptogr., 89 (2021), 1615-1638.  doi: 10.1007/s10623-021-00882-8. [8] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylyshchak and B. Yildiz, Bordered constructions of self- dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004. [9] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7. [10] S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574. [11] S. T. Dougherty, J.-L. Kim and P. Solé, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45. [12] S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Group matrix ring codes and constructions of self-dual codes, Applicable Algebra in Engineering, Communication and Computing, 2021. doi: 10.1007/s00200-021-00504-9. [13] J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbours, Finite Fields Appl., 67 (2020), 101727, 12 pp. doi: 10.1016/j.ffa.2020.101727. [14] J. Gildea, A. Kaya, R. Taylor and B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92.  doi: 10.1016/j.ffa.2018.01.002. [15] T. A. Gulliver and M. Harada, On double circulant doubly-even self-dual $[72, 36, 12]$ codes and their neighbors, Austalas. J. Comb., 40 (2008), 137-144. [16] A. Korban, All known Type I and Type II $[72, 36, 12]$ binary self-dual codes, available online at https://sites.google.com/view/adriankorban/binary-self-dual-codes. [17] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature - Inspired evolutionary algorithms for self-dual codes, arXiv: 2012.12248, in submission. [18] A. Korban, S. Sahinkaya and D. Ustun, New singly and doubly even binary $[72, 36, 12]$ self-dual codes from $M_2(R)G$- group matrix rings, Finite Fields Appl., 76 (2021), Paper No. 101924, 20 pp. doi: 10.1016/j.ffa.2021.101924. [19] A. Korban, S. Sahinkaya and D. Ustun, An application of the virus optimization algorithm to the problem of finding extremal binary self-dual codes, arXiv: 2103.07739, in submission. [20] A. Korban, S. Sahinkaya and D. Ustun, New extremal binary self-dual codes of length 72 from composite group matrix rings and the neighbour method integrated to the virus optimisation algorithm, in submission. [21] A. Korban, S. Sahinkaya and D. Ustun, New type i binary $[72, 36, 12]$ self-dual codes from composite matrices and $R_1$ lifts, Advances in Mathematics of Communications, 2021. doi: 10.3934/amc.2021034. [22] A. Korban, S. Sahinkaya and D. Ustun, Generator matrices for the manuscript "New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm", available online at https://sites.google.com/view/serap-sahinkaya/generator-matrices. [23] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000. [24] M. Shi, D. Huang, L. Sok and P. Solé, Double circulant self-dual and LCD codes over Galios rings, Adv. Math. Commun., 13 (2019), 171-183.  doi: 10.3934/amc.2019011. [25] M. Shi, L. Qian and P. Solé, On self-dual negacirculant codes of index two and four, Des. Codes Cryptogr., 86 (2018), 2485-2494.  doi: 10.1007/s10623-017-0455-0. [26] M. Shi, L. Sok, P. Solé and S. Çalkavur, Self-dual codes and orthogonal matrices over large finite fields, Finite Fields Appl., 54 (2018), 297-314.  doi: 10.1016/j.ffa.2018.08.011.

show all references

##### References:
 [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125. [2] J. R. Cuevas, H. J. Wang, Y. C. Lai and Y. C. Liang, Virus optimization algorithm: A novel metaheuristic for solving continuous optimization problems, The 10th Asia Pacific Industrial Engineering Management System Conference, (2009), 2166–2174. [3] S. T. Dougherty, The neighbor graph of binary self-dual codes, Des. Codes and Cryptogr., 90 (2022), 409-425.  doi: 10.1007/s10623-021-00985-2. [4] S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over frobenius rings, Cryptogr. Commun., 12, (2020), 127–146. doi: 10.1007/s12095-019-00380-8. [5] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Adv. Math. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037. [6] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, New extremal self-dual binary codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbors, Int. J. Inf. Coding Theory, 5 (2018), 211-226. [7] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Composite matrices from group rings, composite $G$-codes and constructions of self-dual codes, Des. Codes Cryptogr., 89 (2021), 1615-1638.  doi: 10.1007/s10623-021-00882-8. [8] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylyshchak and B. Yildiz, Bordered constructions of self- dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004. [9] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7. [10] S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574. [11] S. T. Dougherty, J.-L. Kim and P. Solé, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45. [12] S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Group matrix ring codes and constructions of self-dual codes, Applicable Algebra in Engineering, Communication and Computing, 2021. doi: 10.1007/s00200-021-00504-9. [13] J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbours, Finite Fields Appl., 67 (2020), 101727, 12 pp. doi: 10.1016/j.ffa.2020.101727. [14] J. Gildea, A. Kaya, R. Taylor and B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92.  doi: 10.1016/j.ffa.2018.01.002. [15] T. A. Gulliver and M. Harada, On double circulant doubly-even self-dual $[72, 36, 12]$ codes and their neighbors, Austalas. J. Comb., 40 (2008), 137-144. [16] A. Korban, All known Type I and Type II $[72, 36, 12]$ binary self-dual codes, available online at https://sites.google.com/view/adriankorban/binary-self-dual-codes. [17] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature - Inspired evolutionary algorithms for self-dual codes, arXiv: 2012.12248, in submission. [18] A. Korban, S. Sahinkaya and D. Ustun, New singly and doubly even binary $[72, 36, 12]$ self-dual codes from $M_2(R)G$- group matrix rings, Finite Fields Appl., 76 (2021), Paper No. 101924, 20 pp. doi: 10.1016/j.ffa.2021.101924. [19] A. Korban, S. Sahinkaya and D. Ustun, An application of the virus optimization algorithm to the problem of finding extremal binary self-dual codes, arXiv: 2103.07739, in submission. [20] A. Korban, S. Sahinkaya and D. Ustun, New extremal binary self-dual codes of length 72 from composite group matrix rings and the neighbour method integrated to the virus optimisation algorithm, in submission. [21] A. Korban, S. Sahinkaya and D. Ustun, New type i binary $[72, 36, 12]$ self-dual codes from composite matrices and $R_1$ lifts, Advances in Mathematics of Communications, 2021. doi: 10.3934/amc.2021034. [22] A. Korban, S. Sahinkaya and D. Ustun, Generator matrices for the manuscript "New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm", available online at https://sites.google.com/view/serap-sahinkaya/generator-matrices. [23] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000. [24] M. Shi, D. Huang, L. Sok and P. Solé, Double circulant self-dual and LCD codes over Galios rings, Adv. Math. Commun., 13 (2019), 171-183.  doi: 10.3934/amc.2019011. [25] M. Shi, L. Qian and P. Solé, On self-dual negacirculant codes of index two and four, Des. Codes Cryptogr., 86 (2018), 2485-2494.  doi: 10.1007/s10623-017-0455-0. [26] M. Shi, L. Sok, P. Solé and S. Çalkavur, Self-dual codes and orthogonal matrices over large finite fields, Finite Fields Appl., 54 (2018), 297-314.  doi: 10.1016/j.ffa.2018.08.011.
Flowchart of the Neighbourhood-Virus Optimisation Algorithm
New Type I $[72, 36, 12]$ Codes from $\mathcal{G}_i$
 Generator Matrix $r_{A_1}$ $r_{A_2}$ $r_{A_3}$ $r_{A_4}$ $r_{A_5}$ $r_{A_6}$ $\gamma$ $\beta$ $|Aut(C_i)|$ $\mathcal{G}_1$ $(1, 1, 1, 1, 1, 1)$ $(0, 1, 0, 0, 0, 0)$ $(1, 1, 1, 0, 1, 0)$ $(0, 0, 0, 1, 1, 0)$ $(1, 0, 1, 0, 1, 1)$ $(1, 1, 1, 0, 0, 1)$ $0$ $165$ $72$ $\mathcal{G}_2$ $(1, 0, 0, 0, 1, 1)$ $(0, 0, 0, 0, 0, 1)$ $(0, 1, 0, 1, 0, 0)$ $(0, 1, 0, 0, 1, 1)$ $(1, 1, 1, 1, 1, 0)$ $(1, 0, 0, 1, 0, 1)$ $0$ $315$ $72$ $\mathcal{G}_3$ $(1, 1, 0, 1, 0, 0)$ $(1, 0, 1, 1, 1, 1 )$ $(1, 0, 0, 1, 1, 0)$ $(1, 0, 0, 1, 1, 1)$ $(0, 0, 1, 1, 0, 1 )$ $(0, 0, 0, 1, 1, 1)$ $0$ $255$ $36$ $\mathcal{G}_4$ $(0, 1, 0, 1, 1, 1 )$ $(1, 1, 0, 1, 1, 0)$ $(1, 1, 1, 0, 1, 0 )$ $(1, 0, 0, 1, 1, 1)$ $( 0, 0, 1, 0, 1, 0)$ $(1, 1, 0, 0, 1, 0)$ $0$ $309$ $72$ $\mathcal{G}_5$ $(1, 1, 0, 1, 0, 0)$ $(0, 0, 0, 1, 0, 1 )$ $(1, 1, 0, 1, 0, 0)$ $(1, 0, 1, 0, 0, 1 )$ $(1, 1, 0, 0, 0, 1 )$ $(1, 1, 0, 0, 1, 0)$ $36$ $537$ $72$ $\mathcal{G}_6$ $(1, 0, 0, 0, 0, 0 )$ $(0, 0, 1, 1, 0, 1)$ $(1, 0, 0, 1, 1, 1)$ $(1, 1, 0, 0, 0, 1 )$ $(0, 1, 1, 0, 0, 1 )$ $(1, 0, 0, 1, 1, 0 )$ $0$ $231$ $36$
 Generator Matrix $r_{A_1}$ $r_{A_2}$ $r_{A_3}$ $r_{A_4}$ $r_{A_5}$ $r_{A_6}$ $\gamma$ $\beta$ $|Aut(C_i)|$ $\mathcal{G}_1$ $(1, 1, 1, 1, 1, 1)$ $(0, 1, 0, 0, 0, 0)$ $(1, 1, 1, 0, 1, 0)$ $(0, 0, 0, 1, 1, 0)$ $(1, 0, 1, 0, 1, 1)$ $(1, 1, 1, 0, 0, 1)$ $0$ $165$ $72$ $\mathcal{G}_2$ $(1, 0, 0, 0, 1, 1)$ $(0, 0, 0, 0, 0, 1)$ $(0, 1, 0, 1, 0, 0)$ $(0, 1, 0, 0, 1, 1)$ $(1, 1, 1, 1, 1, 0)$ $(1, 0, 0, 1, 0, 1)$ $0$ $315$ $72$ $\mathcal{G}_3$ $(1, 1, 0, 1, 0, 0)$ $(1, 0, 1, 1, 1, 1 )$ $(1, 0, 0, 1, 1, 0)$ $(1, 0, 0, 1, 1, 1)$ $(0, 0, 1, 1, 0, 1 )$ $(0, 0, 0, 1, 1, 1)$ $0$ $255$ $36$ $\mathcal{G}_4$ $(0, 1, 0, 1, 1, 1 )$ $(1, 1, 0, 1, 1, 0)$ $(1, 1, 1, 0, 1, 0 )$ $(1, 0, 0, 1, 1, 1)$ $( 0, 0, 1, 0, 1, 0)$ $(1, 1, 0, 0, 1, 0)$ $0$ $309$ $72$ $\mathcal{G}_5$ $(1, 1, 0, 1, 0, 0)$ $(0, 0, 0, 1, 0, 1 )$ $(1, 1, 0, 1, 0, 0)$ $(1, 0, 1, 0, 0, 1 )$ $(1, 1, 0, 0, 0, 1 )$ $(1, 1, 0, 0, 1, 0)$ $36$ $537$ $72$ $\mathcal{G}_6$ $(1, 0, 0, 0, 0, 0 )$ $(0, 0, 1, 1, 0, 1)$ $(1, 0, 0, 1, 1, 1)$ $(1, 1, 0, 0, 0, 1 )$ $(0, 1, 1, 0, 0, 1 )$ $(1, 0, 0, 1, 1, 0 )$ $0$ $231$ $36$
 [1] Joe Gildea, Abidin Kaya, Adam Michael Roberts, Rhian Taylor, Alexander Tylyshchak. New self-dual codes from $2 \times 2$ block circulant matrices, group rings and neighbours of neighbours. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021039 [2] Steven T. Dougherty, Joe Gildea, Adrian Korban, Abidin Kaya. Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68. Advances in Mathematics of Communications, 2020, 14 (4) : 677-702. doi: 10.3934/amc.2020037 [3] Adrian Korban, Serap Şahinkaya, Deniz Ustun. A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022033 [4] Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267 [5] Maria Bortos, Joe Gildea, Abidin Kaya, Adrian Korban, Alexander Tylyshchak. New self-dual codes of length 68 from a $2 \times 2$ block matrix construction and group rings. Advances in Mathematics of Communications, 2022, 16 (2) : 269-284. doi: 10.3934/amc.2020111 [6] Joe Gildea, Adrian Korban, Adam M. Roberts, Alexander Tylyshchak. Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022021 [7] Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031 [8] Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023 [9] Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011 [10] Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002 [11] Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077 [12] Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219 [13] Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415 [14] Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229 [15] Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047 [16] Keita Ishizuka, Ken Saito. Construction for both self-dual codes and LCD codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021070 [17] Steven Dougherty, Adrian Korban, Serap Șahinkaya, Deniz Ustun. Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022036 [18] Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251 [19] Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433 [20] Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23

2021 Impact Factor: 1.015