Diagonally cyclic equitable rectangles and cyclotomic orthomorphisms

This thesis deals with two related combinatorial topics, namely cyclotomic orthomorphisms and diagonally cyclic equitable rectangles. An orthomorphism θ of a finite field F is a permutation of the elements of F satisfying θ(x) - x is also a permutation of F. The orthomorphism θ is cyclotomic if θ(x)/x is constant on cosets of a subgroup S of the multiplicative group of F. The index of θ is the index of the subgroup S. Two orthomorphisms θ, ϕ are orthogonal if θ - ϕ is a permutation. We present nearly complete solutions to two open problems relating to cyclotomic orthomorphisms posed by A. B. Evans in 1992. In particular, we show that cyclotomic orthomorphisms exist for almost all plausible indices. Also, sets of pairwise orthogonal cyclotomic orthomorphisms of any possible set of indices exist, provided the field is large enough. An equitable rectangle is an r x c rectangle containing v symbols, where each row and column contains the symbols in the most balanced, or equitable, way possible for the given parameters (r, c, v). Motivated by diagonally cyclic Latin squares, we introduce a new type of equitable rectangle called a diagonally cyclic equitable rectangle (DCER). One advantage of studying DCERs is that we are able to easily generate an equitable rectangle that is orthogonal to any DCER. We characterise which parameters (r, c, v) allow the existence of a DCER

Diagonally cyclic equitable rectangles and cyclotomic orthomorphisms

This thesis deals with two related combinatorial topics, namely cyclotomic orthomorphisms and diagonally cyclic equitable rectangles. An orthomorphism θ of a finite field F is a permutation of the elements of F satisfying θ(x) - x is also a permutation of F. The orthomorphism θ is cyclotomic if θ(x)/x is constant on cosets of a subgroup S of the multiplicative group of F. The index of θ is the index of the subgroup S. Two orthomorphisms θ, ϕ are orthogonal if θ - ϕ is a permutation. We present nearly complete solutions to two open problems relating to cyclotomic orthomorphisms posed by A. B. Evans in 1992. In particular, we show that cyclotomic orthomorphisms exist for almost all plausible indices. Also, sets of pairwise orthogonal cyclotomic orthomorphisms of any possible set of indices exist, provided the field is large enough. An equitable rectangle is an r x c rectangle containing v symbols, where each row and column contains the symbols in the most balanced, or equitable, way possible for the given parameters (r, c, v). Motivated by diagonally cyclic Latin squares, we introduce a new type of equitable rectangle called a diagonally cyclic equitable rectangle (DCER). One advantage of studying DCERs is that we are able to easily generate an equitable rectangle that is orthogonal to any DCER. We characterise which parameters (r, c, v) allow the existence of a DCER