Dr. Fernando Piñero will supervise students on the following topics:
LRCs from Curves: Lopez et. at have found a class of LRCs from the Hermitian curve with positive rate with availability N^(1/3) and locality N^(2/3) where N is the length of the code. Matthews, Morrison and Murphy extended this code construction to the Norm-Trace curve. Students who work on this topic will work on LRCs from similar curves to find other LRCs and study their decoding performance.
Reed-Solomon lifted codes: Reed-Solomon lifted codes were introduced by Guo, Kopparty and Sudan in https://arxiv.org/pdf/1208.5413.pdf
Lifted codes from norm-trace curves: A possible direction to explore at the 2024 REU is to consider lifted codes over other curves, such as the work done by Matthews, Morrison and Murphy in https://arxiv.org/pdf/2307.13183.pdf
Previous work from the REU team includes:
Hermitian Lifted Codes: This work extends the LRC construction from Reed-Solomon codes to other AG curves.
https://arxiv.org/pdf/2006.05558.pdf
Improved dimension of Hermitian Lifted Codes: In this work, REU participants use elementary polynomial division techniques to improve the dimension bounds of Hermitian Lifted Codes.
https://arxiv.org/pdf/2302.01557.pdf
Codes from affine polar Grassmannians: Linear codes are a useful tool to study geometric varieties. Many properties of the Grassmannian can be derived from Grassmann codes. In this project, we study properties of polar Grassmannians using linear codes defined by evaluations of minors of a class of special matrices. We might also explore LRCs and LDPC codes from these varieties.
Previous work done by REU participants include:
Affine Polar Grassman codes: Doel Rivera-Laboy and Fernando Piñero González determined the parameters of a special class of codes derived from the Grassmannian. (Hermitian case: https://arxiv.org/abs/2110.08964 ; symplectic case: https://arxiv.org/abs/2206.01872). This case serves as a base case to understand the parameters of the projective case.
Polar Grassmann codes: Sarah Gregory, Fernando Piñero González, Doel Rivera-Laboy and Lani Southern used the affine case to partition the polar Grassmannian into disjoint affine parts and applied known results to determine the minimum distance in the proyective case. (3,6)-Orthogonal Grassmannian https://arxiv.org/abs/2210.12884
Future directions include determining the minimum distance of the symplectic and Hermitian Grassmannian for the (4,8) case.
Binary Goppa codes and Alternant Codes - Alternant codes are one of the classes of codes from which many of the best known codes are derived. In this project we search for other best possible codes over small fields using the binary Goppa code or the alternant code construction. We also consider graph based codes using these codes as component codes.
Previous work from REU participans includes:
Best Known Linear codes: Jan Carrasquillo, Axel Gómez-Flores and Christopher Soto found a binary Goppa code is also a [240, 21, 104] best known binary code, improving the previously known code of [240,21,99]. https://arxiv.org/abs/2010.07278\
Improving the distance bound of trace Goppa codes. Isabel Byrne, Natalie Dodson, Eric Pabón-Cancel, Fernando Piñero Gonzalez and Ryan Lynch found an improvement of the bound on the minimum distance of a binary Goppa code when the Goppa polynomial g(x) is a trace polynomial. (Preprint: https://arxiv.org/abs/2201.03741, article in Designs, Codes and Cryptography: https://link.springer.com/article/10.1007/s10623-023-01216-6)
Machine Learning decoders for linear codes: In this project we consider decoding of linear codes using Neural network classifiers trained on cleverly chosen sets of error patterns and codewords. Our goal is to complement and improve current decoders for RS, BCH and cyclic codes. This is a new REU project.
Dr. Pamela E. Harris and Dr. Jennifer Elder will present a variety of research projects on the combinatorics of parking functions. This overarching theme will allow participants to study and analyze parking functions by leveraging computational techniques and theory. Faculty will also guide the development of open-source computational tools for analyzing parking functions and their statistics, with time devoted to creating a database of parking functions and their generalizations. Students will meet daily, give regular talks about their findings, attend mini-courses, guest talks, and professional development seminars, and will acquire skills in free software development. Students will learn how to collaborate mathematically, working closely in their teams to write up their research into a paper.
Research bakground includes the following articles:
Interval and l-interval Rational Parking Functions
Tomás Aguilar-Fraga, Jennifer Elder, Rebecca E. Garcia, Pamela E. Harris, Kimberly P. Hadaway, Kimberly J. Harry, Imhotep B. Hogan, Jakeyl Johnson, Jan Kretschmann, Kobe Lawson-Chavanu, J. Carlos Martnez Mori, Casandra D. Monroe, Daniel Quiñonez, Dirk Tolson III, Dwight Anderson Williams II. Submitted.
On flattened parking functions
Jennifer Elder, Pamela E. Harris, Zoe Markman, Izah Tahir, and Amanda Verga
Journal of Integer Sequences, Vol. 26 (2023), Article 23.5.8.
Unit interval parking functions and the r-Fubini numbers
S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden. To appear in La Matematica.
Parking functions, Fubini rankings, and Boolean intervals in the weak order of Sn
Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Mart ́ınez Mori.
To appear in Journal of Combinatorics.