Thomas Pender
Research Interests:
My research is focused in combinatorial design theory. In particular, I am interested in explicit constructions of combinatorial matrices such as
weighing and incidence matrices. I also pursue constructions of sets of sequences with good correlation properties, and seek to understand their
relationships to objects like various difference configurations in finite groups. Computational techniques are often employed in order to facilitate
generating stores of examples as well as gaining structural understanding.
Brief Bio:
I received my Bachelor's degree at the University of Lethbridge in 2020. There I studied under Dr. Hadi Kharaghani as a research assistant.
Continuing under Dr. Kharaghani's tutelage, I subsequently completed the Master's of Science (Mathematics) program at the University of Lethbridge
in 2022. I am now a doctoral student at Simon Fraser University where I study under Dr. Jonathan Jedwab.
Education
Current Program: Graduate student (doctoral, mathematics), Simon Fraser University, Present. Supervisor: Dr. Jonathan Jedwab.
Master's Degree of Science in Mathematics, University of Lethbridge, 2022. Supervisor: Dr. Hadi Kharaghani.
"Weighing Matrices: generalizations and related configurations" (U of L Library).
Bachelor's Degree of Science, University of Lethbridge, 2020.
"Balanced Group Matrices: theory and applications" (U of L Library).
Submitted Papers
Jedwab, J. and T. Pender. "Two Constructions of Quaternary Legendre Pairs of Even Length." (submitted, 2024).
Abstract: We give the first general constructions of even length Legendre pairs: there is a quaternary Legendre pair of length (q-1)/2 for every prime power q congruent to 1 modulo 4, and there is a quaternary Legendre pair of length 2p for ever odd prime p for which 2p-1 is a prime power.
Refereed Publications
For every article listed below, there is a doi link pointing to the published material. A link to the publicly available arXiv versions is also provided
whenever possible. NB: The arXiv versions often differ markedly from the versions accepted for publication.
Kharaghani, H., T. Pender and V. Tonchev. "Optimal Constant Weight Codes Derived from Balanced Generalized Weighing Matrices."
Des. Codes Cryptog. 92, no. 10 (2024): 2791-2799.
Abstract: Balanced generalized weighing matrices are used to construct optimal constant weight codes that are monomially inequivalent to codes derived
from the classical simplex codes. What's more, these codes can be assumed to be generated entirely by omega-shifts of a single codeword where omega is a
primitive element of a Galois field. Additional constant weight codes are derived by projecting onto subgroups of the alphabet sets. These too are shown
to be optimal.
Pender, T.. "On Extremal and Near-Extremal Self-Dual Ternay Codes." Discrete Math. 347, no. 6 (2024): 113968.
Abstract: A computational approach to using plug-in arrays, circulant matrices, and negacirculant matrices in the construction and enumeration of
extremal and near-extremal self-dual ternary codes. Isomorphism classes of such codes obtainable from orthogonal designs of dimensions 2, 4, and 8 are
completely enumerated for several lengths. Additionally, partial searches are conducted for larger lengths, and weight enumerators are derived for
near-extremal codes.
Kharaghani, H., T. Pender, C. Van't Land and V. Zaisev. "Bush-Type Butson Hadamard Matrices." Glas. Mat. 58, no. 2 (2023): 247-257.
Abstract: Bush-type Butson Hadamard matrices are introduced. It is shown that a nonextendable set of mutually unbiased Butson Hadamard matrices is
obtained by adding a specific Butson Hadamard matrix to a set of mutually unbiased Bush-type Butson Hadamard matrices. A class of symmetric Bush-type
Butson Hadamard matrices over the group G of n-th roots of unity is introduced that is also valid over any subgroup of G. The case of Bush-type Butson
Hadamard matrices of even order will be discussed.
Kharaghani, H., T. Pender and Sho Suda. "Quasi-Balanced Weighing Matrices, Signed Strongly Regular Graphs, and Association Schemes."
Finite Fields Appl. 83, no. 25 (2022): 102065.
Abstract: A weighing matrix W is quasi-balanced if |W| |W|^T = |W|^T |W| has at most two off-diagonal entries, where |W|_{i,j} = |W|_{j,i}. A
quasi-balanced weighing matrix W signs a strongly regular graph if |W| coincides with its adjacency matrix. Among other things, signed strongly regular
graphs and their association schemes are presented.
Kharaghani, H., T. Pender and Sho Suda. "Balanced Weighing Matrices." J. Combin. Theory Ser. A 186, no. 18 (2022): 105552.
Abstract: A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight p in a weighing
matrix W(n, p) is a prime power, it is shown that there is a balanced weighing matrix with Ionin-type parameters. Equivalence with certain classes of
association schemes is discussed in detail.
Kharaghani, H., T. Pender and Sho Suda. "A Family of Balanced Weighing Matrices." Combinatorica 42, no. 6 (2022): 881-894.
Abstract: Balanced weighing matrices with parameters [1+18(9^{m+1}-1)/8, 9^{m+1}, 4 9^m] for each nonzero integer m are constructed. This is the first
infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association
scheme.
Kharaghani, H., T. Pender and Sho Suda. "Balancedly Splittable Orthogonal Designs and Equiangular Tight Frames." Des. Codes Cryptog. 89, no. 9,
(2021): 2033-2050.
Abstract: The concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. As an application, equiangular
tight frames over the real, complex, and quaternions meeting the Delsarte-Goethals-Seidel upper bound are obtained.
Source Code
Search for even length quaternary Legendre pairs:
[ tarball ]
[ repository ]
Search for (near-)extremal self-dual ternary codes:
[ tarball ]
[ repository ]
Auxilary scripts for generating various combinatorial matrices:
[ tarball ]
[ repository ]
Search for mutually orthogoval affine translation planes:
[ tarball ]
[ repository ]
Command line indexing utility:
[ tarball ]
[ repository ]
Presentations
Pender T.. "Balanced Weighing Matrices." Presented at the Coast Combinatorics Conference at The University of Victoria, Victoria, BC, November 2023:
[ beamer ]
[ repository ]
[ website ]
Pender T.. "Balancedly Splittable Orthogonal Designs." Presented at the Alberta-Montana Combinatorics and Algorithm Days at
The Banff International Research Station, Banff, AB, June 2022:
[ beamer ]
[ repository ]
[ website ]
Pender T.. "Balanced Generalized Weighing Matrices and Optimal Codes." Presented at the Canadian Mathematical Society's Winter Meeting. December 2021:
[ beamer ]
[ repository ]
[ website ]
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Department of Mathematics
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Email: tsp7@sfu.ca
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Simon Fraser University
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8888 University Drive
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Burnaby BC V5A 1S6
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CANADA
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