Selected Topics in Computing and Informatics /
A Course on Algebraic Coding Theory
Lecturer:
Prof. Michael Lang, Ph.D., Bradley University, Department of Mathematics
Course timing: Oct.-Jan. 2017, 15 weeks, 45 hours
Objectives:
The primary objective is to transmit knowledge and problem-solving skills in the area of coding
theory. This will include background in design theory. We will discuss not only combinatorial
approaches but algebraic ones as well.
The secondary objective is to develop appreciation for the interaction of superficially distinct subjects,
particularly how problems in one area can be translated to another wherein their solutions are more
forthcoming.
Syllabus: The course will include selected advanced topics in:
1. Latin squares (orthogonal arrays, conjugates and isomorphism, partial and incomplete Latin
squares, counting Latin squares, the Evans conjecture)
2. Hadamard matrices, Reed-Muller codes (Hadamard matrices and conference matrices, recursive
constructions, Payley matrices, Williamson's method, excess of a Hadamard matrix, first order
Reed-Muller codes)
3. Designs (the Erdös-De Bruijn theorem, Steiner systems, Hadamard designs, counting, incidence
matrices, the Wilson-Petrenjuk theorem, symmetric designs, projective planes, derived and
residual designs, the Bruck-Ryser-Chowla theorem, constructions of Steiner triple systems, writeonce memories)
4. Codes and designs (terminology of coding theory, the Hamming bound, the Singleton bound,
weight enumerators and MacWilliams’ theorem, the Assmus-Mattson theorem, symmetry codes,
the Golay codes, codes from projective planes)
5. Strongly regular graphs and partial geometries (the Bose-Mesner algebra, eigenvalues, the
integrality conditions, quasisymmetric designs, the Krein condition, the absolute bound, uniqueness
theorems, partial geometries, examples)
6. Orthogonal Latin squares (pairwise orthogonal Latin squares and nets, Euler's conjecture, the BoseParker-Shrikhande theorem, asymptotic existence, orthogonal arrays and transversal designs,
difference methods, orthogonal subsquares)
7. Projective and combinatorial geometries (projective and affine geometries, duality, Pasch's axiom,
Desargues’ theorem, combinatorial geometries, lattices, Greene's theorem)
8. Gaussian numbers and q-analogues (chains in the lattice of subspaces, q-analogue of Sperner's
theorem, interoperation of the coefficients of the Gaussian polynomials, spreads)
Intended learning outcomes:
Knowledge and understanding: The students will understand many combinatorial structures,
particularly from coding and design theory. They will also see how algebra can be tied to these
objects.
Application: The students will be able to solve problems involving the combinatorial structures
discussed and argue that their solutions are correct.
Reflection: Having seen examples where such a technique is productive, the students will look
for opportunities to bring another subject to bear on a problem, making it more tractable.
Transferable skills: The experience with codes and designs will be useful in many technical areas.
Moreover, the increase in problem-solving ability is even more universally applicable.
Prerequisite knowledge: Calculus and modern algebra would be helpful but are not absolutely required.
Literature: J. H. Van Lint and R. M. Wilson, A Course in Combinatorics, (2nd Ed.)
Lecturer's references:
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Lang, M. “Bipartite distance-regular graphs: the Q-polynomial property and pseudo primitive idempotents".
Discrete Mathematics 331 (2014), 27-35.
Lang, M. “Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs".
European Journal of Combinatorics 29 (2008), 34-44.
Lang, M, Terwilliger, P. “Almost-bipartite Q-polynomial distance-regular graphs". European Journal of
Combinatorics 28 (2007), 258-265.
Lang, M. “A new inequality for bipartite distance-regular graphs". Journal of Combinatorial Theory - Series
B 90 (2004), 55-91.
Lang, M. “Leaves in representation diagrams of bipartite distance-regular graphs". Journal of Algebraic
Combinatorics 18 (2003), 245-254.
Lang, M. “Tails of bipartite distance-regular graphs". European Journal of Combinatorics 23 (2002), 10151023.