Summer Research at the University of Richmond

Summer Research at the University of Richmond with Dr. James A. Davis

I have supervised undergraduates in research projects five of the last six summers. Below, you will find pictures of the students involved in these projects together with links to abstracts and papers. Please contact me if you would like further information.

Investigations into a possible new family of Partial Difference Sets. Peter Magliaro and Adam Weaver, summer 2003. We investigated partial difference set constructions for the parameter family $(2^{3e},2^{2e}+2^e+1,2^e+4,2^e+2)$. After working through the simple construction examples, the Fiedler-Klin construction, and the DeLange construction, we attempted to extend these methods to create new partial difference sets. Along with these constructions, we tried some original ideas involving vector spaces and embedding partial difference sets into larger groups. For their report, see Xiang conjecture report. (supported by the National Security Agency grant number MDA904-03-1-0032)
Partial Difference Sets in new groups. Tiffany Broadbent and Ed Kenney, summer 2002. We investigate the connections between partial difference sets and projective planes of several different orders in an attempt to locate a family of partial difference sets in new groups. We firstshow several Galois ring constructions, and then describe work using quadratic forms and Mathon-constructed maximal arcs to provide insight into their geometry and structure. For their report, see Partial Difference Set report. (supported by the University of Richmond Undergraduate Research Committee)
Difference Sets in 2-groups and their associated codes. Mohammed Abouzaid and Jamie Bigelow, summer 2000. Co-directed with Deirdre Smeltzer. Several different construction techniques for difference sets in 2-groups are compared to determine the number of inequivalent difference sets in various groups. Mohammed and Jamie used computer searches as well as Jennings basis computations to determine differences. Report forthcoming! (supported by Hewlett-Packard)
Extended Legendre sequences have good Merit Factor: Tony Kirilusha and Ganesh Narayanaswamy, Summer 1999. Merit factor measures the worth of a binary sequence for application to many different transmission schemes. Tony and Ganesh described many new sequences which have good merit factor: they based these sequences on a known family. They proved some asymptotic results, and they also found some interesting examples from extensive computer searches. For their report, see Merit Factor Report. (supported by Hewlett-Packard)

Intersection of Golay and Kerdock Codes: Mohammed Abouzaid and Nick Gurski, summer 1999. Mohammed and Nick worked on establishing the intersection between two important families of codes. The Golay codes are used for their optimal control of power in a certain transmission scheme, and the Kerdock codes are used for their good error correcting properties. Mohammed and Nick spent much of the summer learning about these two families, and they got some preliminary results on the intersection for large sizes. For their report, see Golay-Kerdock Report. (supported by Hewlett-Packard)
Power properties of Octary Golay Codewords: Katie Nieswand and Kara Wagner, summer 1998. Following the work done by Mike and Meredith the previous summer, Kara and Katie extended their analysis to octary codes (codes which use 8 symbols instead of 2 or 4). Engineers and mathematicians at Hewlett-Packard had been intrigued by computer results for octary codes which did not occur for binary or quaternary codes, and Katie and Kara came up with partial explanations for the data. In the process, they found a proof of why Reed-Muller codes played an important role in codes for this transmission scheme, and one of the leaders of the Coding theory world was intrigued by their observation. For their report, see Octary Report (supported by Hewlett-Packard)
Merit factor of binary sequences: Michael Cammarano and Tony Kirilusha, summer 1998. Merit factor is a measure of how good a sequence for many of the transmission applications. Golay sequences have good power properties, and it seemed natural to examine their merit factors. Mike and Tony explored these sequences and related sequences to see if they could find any asymptotic results on merit factors. They produced a lot of data on small cases, made a number of conjectures about what they thought was happening in the long run, and worked hard to try to push their ideas as far as they could. They got a taste of how difficult mathematics problems can be! For their report, see Merit Factor Report (supported by Hewlett-Packard)
Elliptic Curve Cryptosystems and the Quantum Computer: Jodie Eicher and Yaw Opoku, summer 1997. Moving to a science fiction world, Jodie and Yaw studied applications of a yet-to-be-built quantum computer. If a quantum computer could be built (a machine that operates in the strange world of quantum mechanics rather than classical electronics), that computer would have computational power far exceeding classical machines. In particular, it could ``break'' most modern electronic cryptosystems that enable secure communications. Jodie and Yaw had to learn about elliptic curve cryptography as well as quantum computers to show how to combine them. Their research earned them an award at the Undergraduate Research Symposium at the University of Richmond in April, 1998. For their report, see Elliptic Curve/ Quantum Report (supported by Hewlett-Packard)
Power properties of Quaternary Golay Codewords: Michael Cammarano and Meredith Walker, summer 1997. Wireless communication between computers requires two conditions to be met: first, the transmitted signal needs to have peak power below a certain threshhold, and second, the coding scheme needs to be robust enough to correct the errors that occur in transmission. Under one scheme to transmit codewords, a proposed coding scheme needed additional mathematical analysis. Mike and Meredith proved that this scheme would have several desirable properties, answering some of the questions that had been raised by engineers at Hewlett-Packard Laboratories in Bristol, England. For their report, see Quaternary Report (supported by Hewlett-Packard)
Codes over Rings:Sarah Spence and Brian McKeever, summer 1995. Modern electronic communications has generated a need to understand efficient algorithms for correcting errors which will inevitably occur. Traditionally, coding theorists limited their study to finite fields, most notably binary numbers. Recently, some difficult binary codes were placed in the wider context of codes over rings. Sarah and Brian studied codes over these more general objects. Their research earned them an award in a poster competition at the National Mathematics Meetings in Orlando in January, 1996. No picture available.