**The Database of Z4 Codes **

Check out the Database for Z4 Codes

My primary area of research is Algebraic Coding Theory. I received my PhD
from The Ohio State University.

Here is the abstract of my
PhD thesis from OSU web site.

My PhD advisor was Prof.
Dijen Ray-Chaudhuri , the letter "C" in BCH codes refers to him.

My areas of interest include:

- Algebraic and combinatorial constructions of codes with good parameters over finite fields and rings
- Cyclic codes and their generalizations
- Codes over Z_4 and more general rings
- Code-based cryptography
- Cryptography
- History of Mathematics and Science, especially in the Islamic World. Listen to my story here.
- Relationship between religion and science
- Mathematics Education/Pedagogy
- Theoretical Computer Science

Publications

Books

Miftah al-Hisab, Volume II: Geometry, Translation and Commentary, Springer/Birkhäuser, 2020 (joint work with L. Hammoudi and G. Bakbouk), ISBN 978-3-030-61329-7

Miftah al-Hisab, Volume I: Arithmetic, Translation and Commentary, Springer/Birkhäuser, 2019 (joint work with L. Hammoudi), ISBN 978-3-030-14949-9

Algebraic Coding Theory

· A. S. Karbaski, T. Abualrub, N. Aydin and P. Liu*. Additive Polycyclic Codes over F_4 Induced by Binary Vectors and Some Optimal Codes, accepted for publication in Advances in Mathematics of Communications, 2021.

· N. Aydin, P. Liu*, B. Yoshino*. A Database of Quantum Codes, accepted for publication in Journal of Algebra Combinatorics Discrete Structures and Applications, 2021.

· N. Aydin and O. R. Vandenberg* A New Algorithm for Equivalence of Cyclic Codes and Its Applications, accepted for publication in Applicable Algebra In Engineering, Communication and Computing, 2021,[available online] DOI 10.1007/s00200-021-00525-4

·
A. Dertli, Y. Cengellenmis, and N. Aydin, On Skew Cyclic Codes over a mixed alphabet and their applications to DNA codes, *Discrete Mathematics, Algorithms and Applications, 2021, https://doi.org/10.1142/S1793830921501433*

·
N. Aydin, B. Yildiz, and S. Uludag, A Class of Weak Keys for the QC-MDPC Cryptosystem, *2020 Algebraic and Combinatorial Coding Theory (ACCT)*, Albena, Bulgaria, 2020, pp. 1-4, doi: 10.1109/ACCT51235.2020.9383383.

·
N. Aydin, T. Guidotti*, and P. Liu*, New Linear Codes as Quasi-Twisted Codes from Long Constacyclic Code, *2020 Algebraic and Combinatorial Coding Theory (ACCT)*, Albena, Bulgaria, 2020, pp. 1-6, doi: 10.1109/ACCT51235.2020.9383237.

· N. Benbelkacem, M. F. Ezerman, T. Abualrub, N. Aydin and A. Batoul, Skew Cyclic Codes Over F_4 R, Journal of Algebra and Its Applications. 2250065, (2022), DOI: 10.1142/S0219498822500657

· Y. Cengellenmis, N. Aydin, A. Dertli, Reversible DNA Codes from Skew Cyclic Codes over a Ring of Order 256, Journal of Algebra Combinatorics Discrete Structures and Applications, Vol 8(1), 1-8, 2021.

· T. Abualrub, N. Aydin, I. Aydogdu, Optimal Binary Codes Derived from F2F4-additive Cyclic Codes, Journal of Applied Mathematics and Computing, 64,71–87 (2020). https://doi.org/10.1007/s12190-020-01344-5

· N. Aydin,T Guidotti*, P.Liu*, A Shaikh*, O VandenBerg*, Some Generalizations of the ASR Search Algorithm for Quasi-Twisted Codes, Involve, Vol 13(1), 2020, 137-148.

· A. Melakhessou, N. Aydin, Z. Hebbache,K. Guenda, Zq(Zq+uZq)-linear skew constacyclic codes, Journal of Algebra Combinatorics Discrete Structures and Applications, 17(1), 2020, 85-101.

· N. Aydin,Y. Cengellenmis, A. Dertli,S. T. Dougherty, E. Salturk Skew cyclic and skew constacyclic codes over the local Frobenius non-chain rings of order 16, Advances in Mathematics of Communication, 14(1), Feb 2020, 53-67.

·
N. Aydin, “Some new linear codes from skew cyclic codes and computer algebra challenges”, *Applicable Algebra in Engineering Communication and Computing*, March 2019, https://doi.org/10.1007/s00200-019-00383-1

· N. Aydin, J. Lambrinos*,Oliver VandenBerg*, “On Equivalence of Cyclic Codes, Generalization of a Quasi-Twisted Search Algorithm, and New Linear Codes”, Designs, Codes and Cryptography, 2019, DOI:10.1007/s10623-019-00613-0

· N. Aydin,G. Bakbouk*, J. Lambrinos*, “New Linear Codes over non-Prime Fields”, Cryptography and Communications, 11(5), 891-902, 2019, DOI: 10.1007/s12095-018-0333-2

· N, Aydin, D. Foret*, “New Linear Codes over GF(3), GF(11), and GF(13) ”, Journal of Algebra Combinatorics Discrete Structures and Applications, 6 (1), 13-20, 2019.

· M. Ozen, F. Z. Uzekmek, N, Aydin, N. T. Ozzaim, “One generator quasi-cyclic codes over F2+v ”, J. Appl Math & Informatics, Vol 36, 2018, 359-378, https://doi.org/10.14317/jami.2018.359

· Y. Cengellenmis, A. Dertli, N. Aydin, “Some constacyclic codes over Z4[ u]/ < u^2>, new Gray maps, and new quaternary cod”, Algebra Colloquim, 25(3), 2018, 369-376

· R. Dastbasteh, H. Mousavi, T. Abualrub, N. Aydin, J. Haghighat, “Skew cyclic codes over F_p+uF”, Int. J. of Information and Coding Theory, Vol 5. No. 1, 2018, 81--99.

· N. Aydin, A. Dertli, Y. Cengellenmis, “On some constacyclic codes over Z4[ u]/ < u^2-1>, their Z4 images, and new codes”, Des. Codes Cryptogr. 86(6), 2018, :1249–1255, DOI: 10.1007/s10623-017-0392-y

·
M. Ozen, T. Ozzaim, N. Aydin, “Cyclic codes over Z4 + uZ4 + u^2”, *Turkish J. Math. * 41(5), 2017, 1235-1247, DOI: 10.3906/mat-1602-35

· A. Batoul, K. Guenda, T. A. Gulliver, N. Aydin, “Constacyclic codes over finite principal ideal rin” International Conference on Codes, Cryptology, and Information Security C2SI 2017, Springer LNCS 10194, pp 161-175, DOI 10.1007/978-3-319-55589-8_11

· A. Batoul, K. Guenda, T. A. Gulliver, N. Aydin, “On isodual cyclic codes over finite chain rings” International Conference on Codes, Cryptology, and Information Security C2SI 2017, Springer LNCS 10194, pp 176-194, DOI 10.1007/978-3-319-55589-8_12.

·
N. Aydin, N. Connolly*, J. Murphree* “New binary linear codes from QC codes and an augmentation algorithm”*AAEC. *28(4) 339-350, August 2017, DOI https://doi.org/10.1007/s00200-017-0327-x

·
N. Aydin, A. Halilovic, “ A Generalization of Quasi-twisted Codes: Multi-twisted codes”*, Finite Fields and Their Applications, 45 (2017), 96-106, DOI: http://dx.doi.org/10.1016/j.ffa.2016.12.002*

·
M. Shi, L. Qian, L. Sok, N. Aydin, P. Sole“ On constacyclic codes over Z4[u]/<u^2-1>”, *Finite Fields and Their Applications, 45 (2017), 86–95, DOI: http://dx.doi.org/10.1016/j.ffa.2016.11.016*

· N. Aydin, N. Connolly*, and M. Grassl,“Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes”, Advances in Mathematics of Communication, Vol 11(1), 2017, 245-258, DOI: 10.3934/amc.2017016

*.*

·
T. Abualrub, N. Aydin,“Additive Cyclic Codes over Mixed Alphabets and the Football Pool Problem”*, Discrete Math. Algorithm. Appl.* **09**, 1750010 (2017) [13 pages] DOI: http://dx.doi.org/10.1142/S1793830917500100

· N. Aydin, and T.Abualrub. “Optimal quantum codes from additive skew cyclic codes”, Discrete Mathematics, Algorithms, and Applications, 8(1), 2016 DOI:10.10.1142/S1793830916500373.

· M. Ozen, F. Z. Uzekmek, N, Aydin, N. T. Ozzaim, “Cyclic and some constacylic codes over the ring Z4[u]/<u^2-1> ”, Finite Fields and Their Applications, Vol 38, 2016, 27-39, https://doi.org/10.1016/j.ffa.2015.12.003

· R. K. Bandi, M. Bhaintwal, and N. Aydin. “A mass formula for negacyclic codes of length 2^k and some good negacyclic codes over Z4+uZ4”, Cryptography and Communications, 2016, DOI: 10.1007/s12095-015-0172-3.

· E. Chen and N. Aydin, "A database of linear codes over F13 with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm", JACODESMATH (Journal of Algebra, Combinatorics, Discrete Structures and Applications), Vol 2 (1), 2015,1-16.

· E. Chen and N. Aydin, "New quasi-twisted codes over F11-minimum distance bounds and a new database", J. of Information and Optimization Sciences, Vol 36 (1-2), 2015,129-157.

·
B. Yildiz and N. Aydin,

·
N. Aydin, J. Murphree*

·
S. Karadeniz, B. Yildiz and N. Aydin,

·
T. Abualrub, I. Siap and N. Aydin,

·
N. Aydin, S. Karadeniz, and B. Yildiz,

·
T. Abualrub, N. Aydin, and P. Seneviratne,

·
I. Siap, T. Abualrub, N. Aydin, and P. Seneviratne,

·
I. Siap, T. Abualrub and N. Aydin

·
N. Aydin, and T. A. Gulliver,

·
R. Ackerman*, and N. Aydin,

·
T. Abualrub, A. Ghrayeb,

·
N. Aydin, T. Asamov*,

·

· T. Asamov*, and N. Aydin. “A search algorithm for linear codes: progressive dimension growth”, Designs, Codes and Cryptography, Vol. 45, No 2, pp. 213-217, November 2007.

·
*Proceedings of International Symposium on
Information Theory* (ISIT 2007), June 24-29 2007, Nice, France, 856-860.

·
T. Asamov*,
and N. Aydin, “LDPC codes
of arbitrary girth”, In *Proceedings of the 10 ^{th} Canadian Workshop on
Information Theory *(CWIT 2007), June 6-8 2007, Edmonton, Alberta, Canada, 69-72.

·
N. Aydin and D. Ray-Chaudhuri “On some
classes of optimal and near-optimal polynomial codes”, Finite Fields and Their Applications*.** *Vol.
**10**, No. 1, pp. 24-35,
January 2004.

· N. Aydin and I. Siap “New quasi-cyclic codes over F5” Applied Mathematics Letters, Vol 15, No 7, pp 833-836, October 2002

·
N. Aydin and D. Ray-Chaudhuri “Quasi-cyclic codes over Z4 and some new binary codes” IEEE Transactions on Information Theory** **Vol.
48, No. 7, pp. 2065-2069, July 2002.

·
N. Aydin, **, **Vol.
24, No. 3, pp. 313-326, December 2001**.**

·
I. Siap, N. Aydin and D. Ray-Chaudhuri “New
1-generator quasi-twisted codes over GF(5)” In *Codes*** ***and Association Schemes*** ***(**Piscataway, NJ, 1999**)**,*
pp. 265-275, DIMACS Ser.
Discrete Math. & Theor. Comp. Sci., *AMS.,
*

· I. Siap, N. Aydin and D. Ray-Chaudhuri “New ternary quasi-cyclic codes with improved minimum distances”,IEEE Transaction on Information Theory, Vol. 46, No. 4, pp. 1554-1558, July 2000.

* denotes a Kenyon undergraduate

History of Science

• N. Aydin, "The Role of Religion in the Development of Mathematical Sciences in Medieval Islamic Civilization", Muslim Heritage, 2021.

• N. Aydin and L. Hammoudi, "Root Extraction by Al-Kashi and Stevin", Archive for History of Exact Sciences, Vol 69 (3), pp 291-310, May 2015. DOI 10.1007/s00407-015-0150-3

• Algebra, in *The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam*, May 2014.

• Geometry, in *The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam,* May 2014.

• N. Aydin and M.Rihtim “A striking example of connections between Renaissance Europe and the Muslim World: Copernicus and Tusi” Proceedings of the international conference devoted to the 810th years of Nasireddin Tousi, June 2011, Baku-Azerbaijan, pp. 53-64, National Academy of Sciences of Azerbaijan.

Computer Science

• N. Aydin and P. A. Sivilotti “Remote Belief: Preserving Volition for Loosely-Coupled Processes”, in proceedings of ICDCS 2003, 23 rd International Conference on Distributed Computing Systems, Providence, Rhode Island, pp. 434-440, May 19-22, 2003

Pedagogy/Mathematics Education

• N. Aydin, “Impacts of a unique course on the history of mathematics in the Islamic World”, MAA Convergence, July 2017

• A. S. Ozdemir, E. Guler, N. Aydin, “Effects of Cryptographic Activities on Understanding Modular Arithmetic”, Turkish Journal of Computer and Mathematics Education, Vol.2 No.3, 247-256, 2011.

• N. Aydin “Enhancing Undergraduate Curriculum via Coding Theory and Cryptography”, PRIMUS (Problems, Resources and Issues in mathematic undergraduate studies) Volume 19 Issue 3, 296-309, May 2009.

• N. Aydin and E. Halat “The impacts of undergraduate mathematics courses on college students' geometric reasoning stages”, Montana Mathematics Enthusiasts, Vol 6, Nos 1&2, 151-164, 2009.

• E. Halat, E. Jakubowski E. and N. Aydin “Reform-Based Curriculum and Motivation in Geometry”, Eurasia Journal of Mathematics, Science and Technology Education, Vol 4, No 3, 285-292, 2008.

• N. Aydin “Elliptic Curve Cryptography”, A Computational Science Module developed as part of an NSF grant. December 2011.

• N. Aydin “Public Key Cryptography and the RSA Cryptosystem”, A Computational Science Module developed as part of an NSF grant. July 2009.

• N. Aydin “An Introduction to Coding Theory via Hamming Codes”, A Computational Science Module developed as part of an NSF grant. August 2007.

Conference Presentations

·
“New Linear Codes as QT codes from Long Constacyclic Codes”, ACCT 2020 (17th Int Workshop on Algebraic and Combinatorial Coding Theory), Bulgaria, October 2020 (N. Aydin*,** T. Guidotti and P. Liu**).

· “ Al-Kashi's Miftah al-Hisab: a 21st century translation and reading”, Joint Mathematics Meetings 2020, in AMS Special Session on Current Areas of Interest in the Mathematical Sciences of Medieval Islam, Denver, CO, USA January 2020 (N Aydin and L. Hammodi*).

·
“A New Algorithm for Equivalence of Cyclic Codes and Its Applications”, AMS Contributed Paper Session on Computer Science, Programming, Information and Related Topics, Joint Mathematics Meetings 2020, in, Denver, CO, USA January 2020 (N Aydin, and **R. O. VandenBerg***).

·
“New Results on Equivalence of Cyclic Codes and a Generalization of a Search Algorithm", Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, FL, March 2019 (N. Aydin and **R. O. VandenBerg***)

· “Code-Based Crytography for the Internet-of-Things", 4 hour tutorial in IEEE Consumer Communications & Networking Conference, Las Vegas, NV, January 2019 (N. Aydin, S. Uludag, and B. Yildiz).

· “A Generalization of Quasi-Twisted Codes: Muti-twisted Codes", AMS Sectional Meeting (#1136), The Ohio StateUniversity, Columbus, OH, March 2018.

· “ Translation of Miftah al-Hisab", Midwest History of Math Conference and ORESME Joint Meeting, Wabash College, Crawfordsville, IN, Sep 2017.

· “ Recent Methods of Constructing New Linear Codes over Z4”, Mathematical Congress of the Americas, in Special Session on Finite Algebraic Combinatorics and Applications, Montreal, Canada, July 2017.

·
“ Legacy of Al-Kashi: Why does it matter in the 21st century”, Joint Mathematics Meetings 2016, in AMS Special Session on Current Areas of Interest in the Mathematical Sciences of Medieval Islam, Seattle, WA, USA January 2016 (**N. Aydin***, and L. Hammoudi)

· “ New Binary Linear Codes from Quasi-Cyclic Codes and an Augmentation Algorithm", The International Conference on Coding Theory and Cryptography (ICCC 2015), USTHB, Algiers, Algeria, November 2015.

· “ The Legacy of Islamic Science in the Modern World: A Few Examples”, Grand Dialogue in Religion & Science, Grand Valley State University, Grand Rapids, MI, June 2015.

· “Computer Algebra Challenges for Constructing Skew Cyclic Codes”, ACA (Applications of Computer Algebra) 2014, Fordham University, NY, July 2014.

· “New Databases of Linear Codes over GF(11) and GF(13)”, Karatekin Mathematics Days, at Cankiri Karatekin University, Turkey, June 2014.

· “A History of Mathematics Course for the Liberal Arts”, MAA-Ohio Section Spring 2014 meeting, University of Toledo, Toledo, OH, April 2014.

·
“Islam, History and Mathematical Sciences”, Online lecture,

·
“Recent Results in Skew Cyclic Codes”, 31st Ohio State-Denison Mathematics Conference,

·
“Skew Additive Cyclic Codes", ICAAA 2011, International Conference on Applied Analysis and Algebra,

· “A Striking Example to Connections Between Europe and the Muslim World: Copernicus and Tusi” International conference on the history, philosophy of science and technology devoted to the 810-th anniversary of Nasireddin Tousi, June 2011, Baku, Azerbaijan

·
“A Database of Z4 Codes”, The first Int. Conference on Mathematics and
Statistics,

·
“Search for Good Linear Codes over Z_4 and
a New Database”, International Conference on Algebra and its
Applications,

·
“Search for Good Linear Codes”, 29th
OSU-Denison Conference, Special Session in Combinatorics, The

·
“Some open problems on quasi-twisted and
related code constructions and good quaternary codes”, International
Symposium on Information Theory,

·
“Enhancing undergraduate mathematics
curriculum via coding theory and cryptography”, Poster presentation, in 3^{rd}
International Conference on the Teaching of Mathematics at the Undergraduate
Level, Istanbul, Turkey, July 2006.

· “Enhancing undergraduate mathematics curriculum via coding theory and cryptography”, MAA Spring 06 meeting, in Ohio Project Next Section, University of Akron, Akron, OH, March 2006.

·
“Affine Cyclic Codes”, AMS Sectional
Meeting,

·
“On a Few Classes
of Optimal and Near-Optimal Polynomial Codes”, MTNS 2002,

· “A Class of Near Optimal Polynomial Codes”, Joint Mathematics Meetings, AMS Special Session in Algebraic Coding Theory, San Diego, CA. January 2002.

·
“A Family Polynomial Codes and a Class of
Griesmer Codes”, AMS Sectional Meeting, Special Session in Codes and
Designs, The

· “The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes” XVth OSU-Dennison Conference, Special Session in Combinatorics, The Ohio-State University, OH, May 2000.

·
“Quaternary Quasi-Cyclic Codes and Some New
Binary Codes”

·
“New Ternary Quasi-Cyclic Codes with
Improved Minimum Distances” AMS Sectional
Meeting,

Other Presentations

· “A Mathematician's Journey into History of Mathematics: Impacts, Reflections, and Lessons", invited national talk, MAA Special Interest Group on History of Mathematics (HOM SIGMAA), November 3, 2021 (recording available from the presenter).

· “Islamic Art and Geometric Designs of Mosques", presentation in the workshop titled Exploring the Sacred in the Islamic Tradition organized by Spirituality Network, Columbus, OH November, 2019

· "Research in Coding Theory: An Efficient Way to Mathematical Fame and a Promise for the Quantum Age", Colloquim Talk, New College of Florida, Sarasota, FL, March 2019.

· “ The Legacy of Islamic Math and Science: Why does it Matter in the 21st Century?", Colloquim Talk, New College of Florida, Sarasota, FL, March 2019.

· “ The Legacy of Islamic Math and Science: Why does it Matter in the 21st Century?", Mathematics Colloquim, Northern Arizona University, Flagstaff, AZ, March 2018.

· “Recent Results in Quasi-Twisted and Related Codes, and Codes over Rings", ACGT Seminar, Northern Arizona University, Flagstaff, AZ, March 2018.

· “Teaching a More Accurate and Inclusive History of Science", (poster) Franklin University Scholars Showcase, Columbus, OH, October 2018.

·
“The Legacy of Islamic Math and Science: A Few Examples”, Oberlin College, Oberlin, OH,

·
“The Legacy of Islamic Math and Science: A Few Examples”, University of Michigan-Dearborn, Dearborn, MI,

·
“The Legacy of Islamic Science in the Modern World: A Few Examples”, Rockford University, Rockford, IL,

·
“Islam & Science: The Legacy of Islamic Science in the Modern World, Noor Islamic Cultural and Community Center,

·
“A Perfect Introduction to Coding Theory and Search for Good Codes”, Oberlin College, Oberlin, OH,

· “Some generalizations of cyclic codes: Consta-cyclic, quasi-cyclic and quasi-twisted codes”, lecture given in IMA Summer School, at University of Notre Dame, South Bend, IN, June 2004.