Topics in coding theory and combinatorial structures
In this dissertation we study some problems in the theory of error-correcting codes and combinatorics.
In the first Chapter we introduce some facts from elementary number theory, abstract algebra, Galois theory and algebraic structures, including: groups, permutation groups, finite fields, characters of Abelian groups, and polynomials, that will be used in later chapters. Additionally, there is brief overview of combinatorial theory, such as: designs, finite geometry, and the theory of error-correcting codes that we need for our work.
In Chapter 2, we give an algorithm for generating cyclic self-orthogonal codes; the number of cyclic self-orthogonal and maximal cyclic self-orthogonal q-ary of length n, for an arbitrary positive integer n with gcd(n, q) = 1 has been determined. A dimension formula for cyclic self-orthogonal q-ary codes of length qn − 1 with maximal dimension has been given. In Section 2.4, a mass formula and a classification algorithm for cyclic self-orthogonal q-ary codes of length n with maximal dimension is given. In Section 2.5 we classify cyclic self-orthogonal codes with maximum dimension.
In the Chapter 3, we deal with quasi-symmetric designs. The main result of this chapter is that every cyclic quasi-symmetric 2-(63, 15, 35) design with intersection numbers x = 3 and y = 7 is isomorphic to the quasi-symmetric 2-(63, 15, 35) design defined by the 3-dimensional subspaces in PG(5, 2). The method applied in the chapter can be used to generate and classify other cyclic self-orthogonal designs.
In chapter 4, we give an algorithm to list and classify generalized Hadamard matrices of a given order over an arbitrary elementary Abelian group. Generalized Hadamard matrices of order ≤12 over Abelian groups [special characters omitted], and [special characters omitted] have been classified up to equivalence. We have shown that generalized Hadamard matrices of order 4, 8, and 12 over EA(4) are unique up to equivalence. We have computed the Kronecker sum of unique matrices of order 4, 8 and 12 by 226 inequivalent generalized Hadamard matrices of order 16. We have shown that the resulting matrices yield single error-detecting or single error-correcting codes.
In the last Chapter 4, we introduce some open problems related to previous chapters. All computations described in this document have been implemented using Magma or Mathematica. The source codes of these aforementioned computations can be found in the appendices.