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 FiedlerKlin 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 MDA9040310032)

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 Mathonconstructed 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 2groups and
their associated codes. Mohammed Abouzaid and
Jamie Bigelow, summer 2000. Codirected with Deirdre Smeltzer.
Several different construction techniques for difference sets in 2groups
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 HewlettPackard)

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 HewlettPackard)


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 GolayKerdock
Report. (supported by HewlettPackard)

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 HewlettPackard
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 ReedMuller
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 HewlettPackard)

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 HewlettPackard)

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 yettobebuilt 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 HewlettPackard)

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 HewlettPackard Laboratories in Bristol,
England. For their report, see Quaternary
Report (supported by HewlettPackard)

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.