Undergraduate Research

There are many problems and projects in Algebraic Coding Theory that are accessible to undergraduates. In general, we combine algebraic theory with programming skills to search for new codes, codes with better parameters than currently best known codes. This is one of the main problems of coding theory. One of the interesting feature of the problem is that it can be approached from many different perspectives using a variety of tools, some very sophisticated (e.g. advanced algebraic geometry), some rather elementary (e.g. elementary but insightful search algorithms). If you are interested in doing research in this area I encourage you to contact me. Projects are available for beginners and experts. Below is a list of undergradute projects, papers and presentations produced by Kenyon students. Minimal background needed to get started is a course on Linear Algebra and some programming experience. Additionally, courses on Abstract Algebra, Combinatorics, and advanced programming (e.g. Data Structures) are highly recommended.

In addition to our active undergraduate research program with Kenyon students, I also mentor highly motivated and exceptionally strong international high school students for research in coding theory and cryptography offered by Pioneer Academics. This unique program connects talented high school students from all around the world with resources to conduct research. We have also published papers with Pioneer scholars such as this.

Research Projects with Kenyon Students

•     Trang Nguyen '26: Elementary Constructions of Best Known Quantum Codes, Summer 2024

•     Long Tran '26: Elementary Constructions of Best Known Quantum Codes, Summer 2024

•     Moe Belgith'26: Cyclic codes of even length over Z4, Summer 2024.

•     Godwin Idowu'27: (Cascade Scholar): Cyclic codes of even length over Z4, Summer 2024.

•     Yiyang (Lucas) Lu'23: Factorization in skew polynomial rings, Scmp Capstone Project, Spring 2023.

•     Saurav Pandey'23: Codes over GF(17) and GF(19), independent research, Fall 2022-Spring 2023.

•     Yiyang (Lucas) Lu'23: Z4 Codes + Developing an API involving code simulations, Summer 2022

•     Fatma Mahmoud '25: BCH Searches, Summer 2022

•     Vishad Onta'23: ASR search for Z4 Codes and an updated database of Z4 Codes, Summer 2022, preprint available in arxiv.

•     Dev Akre'23 (Cascade Scholar): Generalized CC equivalence, Summer 2021, preprint available in arxiv.

•     Saurav Pandey'23: 2-gen QT codes, Summer 2021, preprint available in arxiv.

•     Matt Harrington'22: ConstX and QC codes, making codes great again, Summer 2021, preprint available in arxiv.

•     Bryan Yoshino'21 and Paul Liu: A Database of Quantum Codes, Spring/Summer 2021, preprint available in arxiv. (accepted for publication in J. Algebra Combinatorics Discrete Structures and Applications)

•     Lynn Butzlaff'22: Special Project in Digital Storytelling about Dissemination of the Research Findings on Contributions of the Medieval Islamic Civilization to Mathematics and Sciences, Summer 2020. Youtube links.

•     Bryan Yoshino'21: Polycyclic codes associated with trinomials, Summer 2020, preprint available in arxiv

•     Thomas Guidotti'21: Top-down search method, ICY-method, Summer 2020,

•     Paul Liu'22: Top-down search method, ICY-method, Polycyclic codes, Summer 2020. Additive polycyclic codes over F4, Summer 2021, preprint available in arxiv.

•     Thomas Guidotti'21: A Generalization of ASR algorithm, MT codes, Summer 2019, preprint available in arxiv.

•     Paul Liu'22: A Generalization of ASR algorithm, MT codes, Summer 2019.

•     Armiya Shaikh'21 (Cascade scholar): A Generalization of ASR algorithm, a greedy algorithm for linear code search, Summer 2019.

•     Ghada Bakbouk'19 : Skew cyclic and skew constacylic codes, Scmp capstone project, Sp 2019.

•     Ghada Bakbouk'19 : Translation of Miftah al-Hisab, Summer 2018, Fall 2018.

•     Oliver Vandenberg '20 : A Generalization of Linear Codes Search Algorithm, Code Equivalence, Code-Based Cryptography, Summer 2018, Fall 2018, Sp 2019, preprint available in arxiv (accepted for publication in AAECC)

•     Jonathan Lambrinos '20: A Generalization of Linear Codes Search Algorithm, Multi-Twisted Codes, Summer 2018.

•     Ghada Bakbouk'19 and Jonathan Lambrinos '20 (Cascade Scholar): New Linear Codes over Non-Prime Fields, Summer 2017.

•     Derek Foret '17: Methods of Construction of New Linear Codes from QT Codes, Summer 2015.

•     John Murphree'15 and Nick Connolly'15: QC Search over the Binary Field, Spring 2015.

•     Nick Connolly'15: Constructing New Linear Codes with Record Breaking Parameters over GF(7), Summer 2014.

•     John Murphree'15: New Linear Codes from Constacyclic codes, Summer 2013.

•     Ryan Ackerman'10: QT codes over GF(5), Summer 2009.

•     Tsvetan Asamov '08: Low Density Parity Check Codes II, Summer 2006

•     Tsvetan Asamov '08: Methods of Constructions for Low Density Parity Check Codes, Summer 2005

•     Irina Ivan '08: Low Density Parity Check Codes, Summer 2005

•     Ben Johnson '06: Multi-twisted Codes, Summer 2004

Publications with Kenyon Undergraduates

· H. ·  Ou-azzou, M. Najmeddine, N. Aydin, P. Liu*, B. Ialou, E. M. Mouloua, Linear codes invariant under a linear endomorphism, accepted for publication in Advances in Mathematics of Communication, 2024.

·     S. R. Pandey*, N. Aydin, E. Z. Chen, F. Jönsson, K. Klonowska, New Results and Bounds on Codes over GF(19), accepted for publication in J. Information and Optimization Sciences, 2024

·      T. Abualrub, S. Karbaski, N. Aydin and P. Liu*. Additive polycyclic codes over F4 induced by nonbinary polynomials, accepted for publication in Journal of Applied Mathematics and Computing, 2023.

·         N. Aydin, T. Guidotti*, P. Liu*,  Good classical and quantum codes from multi-twisted codes,  Contemporary Mathematics, Algebra and Coding Theory, 785:7 2023, https://doi.org/10.1090/conm/785/15771

·         A. Bellil, K. Guenda, N. Aydin, P. Liu*, T. Aaron Gulliver. Constacyclic and quasi-twisted codes over Zq[u]/< u^2-1> and new Z4-linear codes. Advances in Mathematics of Communications, 2023 doi: 10.3934/amc.2023026

·         N. Aydin, Y. Lu*, V. Onta*, An updated database of Z4 codes and an Open Problem about Quasi-Cyclic Codes, In: El Hajji, S., Mesnager, S., Souidi, E.M. (eds) Codes, Cryptology and Information Security. C2SI 2023. Lecture Notes in Computer Science, vol 13874. Springer, Cham. https://doi.org/10.1007/978-3-031-33017-9_24

·         F. Caliskan, R. Aksoy, N. Aydin, P. Liu*, Entanglement-Assisted Binary Quantum Codes from Skew Cyclic Codes over F_2 x (F_2+vF_2), Quantum Information Processing, 22, 208 (2023). https://doi.org/10.1007/s11128-023-03945-yin

·         B. Caliskan, N. Aydin, P. Liu*, Skew cyclic codes over Z4+uZ4+vZ4,  Cryptography and Communications, 2023,15 , 845–858, DOI: 10.1007/s12095-023-00645-3

·     D. Akre*, N. Aydin, M. Harrington*, S. Pandey*, A Generalization of  Cyclic  Code Equivalence Algorithm to Constacyclic Codes,  Des. Codes Cryptogr. 91, 763–777 (2023). https://doi.org/10.1007/s10623-022-01124-1

·     D. Akre*, N. Aydin, M. Harrington*, S. Pandey*, New Binary and Ternary Quasi-Cyclic Codes with Good Properties,      Computational and Applied Mathematics, 42(102), 2023. https://doi.org/10.1007/s40314-022-01946-8

·     S. Pandey*, N. Aydin, M. Harrington*, D. Akre*, A Generalization of the ASR Search Algorithm to 2-Generator Quasi-Twisted Codes, ISIT 2022.

·     N. Aydin, P. Liu*, B. Yoshino*. Polycyclic Codes Associated with Trinomials: Good Codes and Open Question, Designs,Codes, and Cryptography 2022.

·     A. S. Karbaski, T. Abualrub, N. Aydin and P. Liu*. Additive Polycyclic Codes over F_4 Induced by Binary Vectors and Some Optimal Codes, Advances in Mathematics of Communications, 2022.

·     N. Aydin, P. Liu*, B. Yoshino*. A Database of Quantum Codes, J. 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, Applicable Algebra In Engineering, Communication and Computing, 2021, DOI 10.1007/s00200-021-00525-

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

·    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.

·     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, Sep 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. 

·      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

·      N. Aydin , N. Connolly*, and J. Murphree*, “New binary linear codes from QC codes and an augmentation algorithm”, Applicable Algebra in Engineering Communication and Computing, 28(4) 339- 350, August 2017, DOI 10.1007/s00200-017-0327-x

·      N. Aydin , and J. Murphree*, “New linear codes from constacyclic codes”,Journal of the Franklin Institute, Vol 351 (3), March 2014, 1691-1699.

·      R. Ackerman* , and N. Aydin, “New quinary linear codes from quasi-twisted codes and their duals” Applied Mathematics Letters, Vol 24, No 4, pp 512-515, April 2011.

·      N. Aydin, T. Asamov*,“ Search for good linear codes in the class of quasi-cyclic and related codes ” Selected Topics in Information and Coding Theory", Edited Book, I. Woungang, S. Misra, S. Chandra Misra (Eds.), Series on Coding and Cryptology, World Scientific Publishing, ISBN:978-981-283-716-5/981-283-716-7, March 2010, 724 pages

·        N. Aydin , and T. Asamov*, “A Database of Z4 Codes”,  J. of Combinatorics, Information & System Sciences (JCISS), Vol. 34 No. 1-4 Comb, 2009, p: 1-12.

·        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.

·         N Aydin , T.  Asamov*, and T. A. Gulliver  “Some open problems on quasi-twisted and related code constructions and good quaternary codes”, In Proceedings of International Symposium on Information Theory (ISIT 2007), June 24-29 2007, Nice, France, 856-860, DOI: 10.1109/ISIT.2007.4557331

·        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, DOI: 10.1109/CWIT.2007.375703

* denotes a Kenyon undergraduate

Conference Presentations by Kenyon Students (or presentations of joint work by Professor Aydin)

·        “An Updated Database of Z4 codes and an Open Problem About QC codes," Codes, Cryptology, and Information Security, C2S1-2023, May 2023,      N.Aydin*, Y. Lu*, V. Onta*).   

·        “Search for New Linear Codes Through the BCH bound and Its Generalizations ” JMM 2023, January 2023, poster presentation by F. Mahmoud*.   

·        “A Generalization of the ASR Search Algorithm to 2-Generator Quasi-Twisted Codes” Joint Mathematics Meetings (virtual), April 2022 (N. Aydin, D. Akre*, M. Harrington*, and S. Pandey*).

·        “A Generalization of the ASR Search Algorithm to 2-Generator Quasi-Twisted Codes” Joint Mathematics Meetings (virtual), April 2022 (N. Aydin, D. Akre*, M. Harrington*, and S. Pandey*).

·        “New Binary and Ternary Quasi-Cyclic Codes with Good Properties” Joint Mathematics Meetings (virtual), April 2022 (N. Aydin, D. Akre*, M. Harrington*, and S. Pandey*).

·        “New Linear Codes from 2-Generator Quasi-Twisted Codes” Joint Mathematics Meetings (virtual), April 2022 (poster presented by S. Pandey*).

·        “Good Codes from Quasi-Cyclic Codes Using ConstructionX” Joint Mathematics Meetings (virtual), April 2022 (poster presented by M. Harrington*).

·        “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*).

·        “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 R. O. VandenBerg*).

·        “Generalization of a Linear Code Search Algorithm”, MAA Undergraduate Student Poster Session, Joint Mathematics Meetings 2020, in, Denver, CO, USA, January 2020 (poster presented by R. O. VandenBerg*). Received Honorable Mention.

·        “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*)

·      "Generalization of a Linear Code Search Algorithm",  poster presentation by Oliver VandenBerg*, Ohio College Summer Research Symposium, Ohio Wesleyan University, July 2018. A video recording of the presentation.

·      "New Linear Codes over Non-Prime Fields",  poster presentation by J. Lambrinos* and G. Bakbouk*, Young Mathematicians Conference, The Ohio State University, August 2017.

·      "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. ( N. Aydin, N. Connolly*, and J. Murphree*)

·      "New Linear Codes from Quasi-twisted codes",  poster presentation by D. Foret*, Young Mathematicians Conference, The Ohio State University, August 2015.

·      "Constructing New Linear Codes from Quasi-twisted Codes over GF(3),GF(11) and GF(13)",   by D. Foret*, MAA MathFest, Washington DC, August 2015.

·      "Constructing New Linear Codes with Record Breaking Parameters",   by N. Connolly*, MAA MathFest, Portland, Oregon,, August 2014.

·      "New Linear Codes from Constacyclic codes",  poster presentation by J. Murphree*, Young Mathematicians Conference, The Ohio State University, August 2013

·       "A Database of Z4  Codes",  by N. Aydin (contributions by T. Asamov*), The first Int. Conference on Mathematics and Statistics, American University of  Sharjah, Sharjah, UAE. March 2010.

·        "Some open problems on quasi-twisted and related code constructions and good quaternary codes”, by N. Aydin (contributions by T. Asamov*), International Symposium on Information Theory,  Nice, France, June 2007.

·       "LDPC Codes of Arbitrary Girth”, by T. Asamov*, AMS Special Session on Algebraic Coding Theory, U of Cincinnati, September 2006.

·       "Search for New Error Correcting Codes" by B. Johnson*, Young Mathematicians Conference, The Ohio State University, August 2004.

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