• Course code:63536B
  • Contents

A Course on Algebraic Coding Theory

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, write-once 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 Bose-Parker-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)
 

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