Gretchen L. Matthews

Gretchen L. Matthews
Associate Professor
Department of Mathematical Sciences
College of Engineering and Science
Clemson University
Clemson, SC 29634-0975
email: gmatthe@clemson.edu

Research interests
My research is in applications of algebraic geometry to coding theory. I study the construction, analysis, and decoding of algebraic geometry codes and parity-check codes (for instance, LDPC codes) in addition to related algebraic structures.

Current projects
Current projects include:

analysis of iterative decoding algorithms for binary and nonbinary parity-check codes via tools from discrete geometry
explicit construction and analysis of algebraic geometry codes from maximal function fields and related families
explicit construction of small-bias sets via Riemann-Roch spaces and related algebraic geometry codes

Research highlights
Our work on algebraic geometry codes

demonstrates that multipoint codes may have better parameters (meaning better error-correcting capabilities with increased efficiency) than comparable one-point codes, suggesting that the study of algebraic geometry codes not be restricted to the one-point case.
provides a minimum distance decoding algorithm for multipoint codes that capitalizes on recent improvements in list decoding algorithms.
illustrates how Riemann-Roch spaces of nonrational places may be utilized in code constructions; in some instances, this leads to codes with double the error-correcting capability of comparable one-point Hermitian codes.
yields some best-known codes that are included in Brouwer's tables.

Our work on analysis of parity-check codes

proves that the generating function of the pseudocodewords of a binary parity-check code is a rational function (a fact proven concurrently and independently by W.-C.W. Li, M. Lu, and C. Wang).
provides algorithms for producing this generating function.
produces short rational functions that generate the irreducible pseudocodewords of binary and ternary parity-check codes; this allows one to study the impact of parity-check matrix choice in code representation.

Our work on structures related to algebraic geometry codes

yields explicit bases for Riemann-Roch spaces of large families of function fields defined by linearized polynomials (including Hermitian and extended norm-trace function fields); these bases may be applied to the construction of small-bias sets and low-discrepancy sequences.
develops and provides computational tools for producing the minimal generating set of a Weierstrass semigroup of a m-tuple of places of an algebraic function field.

Funding
The work above is supported by the following
2009-2012: NSF, Algebraic analysis of parity check codes and iterative decoding, PI, $120,000.
2007-2009: NSA, Codes from algebraic geometry: constructions and algorithms for implementation, PI, $30,000.
2006-2008: NSA, Algebraic geometry codes and related structures, PI, $30,000.
2002-2006: NSF, Applications of semigroups to algebraic geometry codes, PI, $104,837.
2002-2003: ORAU, Semigroups and error-correcting codes, PI, $5000.

Publications

Students
Doctoral students

S. Anderson, (in progress).
J. Peachey, “Explicit bases for Riemann-Roch spaces over function fields with many rational places and applications,” (Ph.D., Mathematical Sciences, expected December 2011).
W. Kositwattanarerk, “Pseudocodewords of parity-check codes,” (Ph.D., Mathematical Sciences, August 2011).
N. Drake, “Decoding of multipoint algebraic geometry codes via lists,” (Ph.D., Mathematical Sciences, December 2009).

Masters students

J. Hyde-Volpe,“Quantum codes from two-point Hermitian codes,” (M.S., Mathematical Sciences, August 2010).
J. Peachey, “On Weierstrass semigroups of some m-tuples on norm-trace curves,” (M.S., Mathematical Sciences, May 2009).
B. Hicks, “Investigating the regularity of decomposition graphs of prisms,” (M.S., Mathematical Sciences, May 2009).
R. Thomas, “Gene networks modeled by polynomials over finite fields,” (M.S., Mathematical Sciences, May 2008).
J. Marshall, “On the number of Weierstrass semigroups of triples on the Hermitian curve,” (M.S., Mathematical Sciences, May 2007).
M. Coleman, “Semigroups and exact minimum distances of codes from a quotient of the Hermitian curve,” (M.S., Mathematical Sciences, May 2005).
S. Graham, “Decoding arrays for two-point codes,” (M.S., Mathematical Sciences, May 2005).
N. Drake, “Exact minimum distances of some two-point Hermitian codes,” (M.S., Mathematical Sciences, May 2004).
T. Michel, “One-point codes using places of higher degree,” (M.S., Mathematical Sciences, May 2004).
K. Durham, “Some Weierstrass semigroups on certain maximal curves,” (M.S., Mathematical Sciences, May 2003).
T. A. Bedford, “Z4-linear codes,” (M.S., Mathematical Sciences, August 2001).

Honors students

J. Hyde-Volpe, “Quantum codes from two-point Hermitian codes,” (B.S., Mathematical Sciences with honors, May 2009).
C. Baber, “Distance 2 colorings of certain generalized Petersen graphs,” (B.S., Mathematical Sciences with honors, May 2007).
R. Robinson, “On the dual and Lipman chains of a special family of numerical semigroups,”(B.S., Mathematical Sciences with honors, May 2004).
J. Bayless, “On the group generated by an n-cycle and an involution,” (B.S., Mathematical Sciences with honors, May 2003).