Quasi-orthogonal cocycles, optimal sequences and a conjecture of Littlewood (original) (raw)
Related papers
2019
We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {± 1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.
C O ] 2 6 A ug 2 01 9 On quasi-orthogonal cocycles
2019
We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.
On The Adjacency-Type Sequences
2017
This paper develops properties of the adjacency-type sequences defined from the adjacency matrix of the Cayley diagram for the direct product of two cyclic groups of order n and m. Also, we study the adjacency-type sequences modulo α from the generating matrices of the adjacency-type sequences when read modulo α. Then we derive the relationships among the orders of the obtained cyclic groups and the periods of the adjacency-type sequences modulo α. Furthermore, we extend the adjacency-type sequences to groups and then we obtain the period of the adjacency-type sequence in the quaternion group Q8.
Almost supplementary difference sets and quaternary sequences
2019
We introduce almost supplementary difference sets (ASDS). For odd mmm, certain ASDS in mathbbZm{\mathbb Z}_mmathbbZm that have amicable incidence matrices are equivalent to quaternary sequences of odd length mmm with optimal autocorrelation. As one consequence, if 2m−12m-12m−1 is a prime power, or mequiv1mod4m \equiv 1 \mod 4mequiv1mod4 is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology.
On Sequences in Cyclic Groups with Distinct Partial Sums
The Electronic Journal of Combinatorics
A subset of an abelian group is sequenceable if there is an ordering (x1,ldots,xk)(x_1, \ldots, x_k)(x1,ldots,xk) of its elements such that the partial sums (y0,y1,ldots,yk)(y_0, y_1, \ldots, y_k)(y0,y1,ldots,yk), given by y0=0y_0 = 0y0=0 and yi=sumj=1ixjy_i = \sum_{j=1}^i x_jyi=sumj=1ixj for 1leqileqk1 \leq i \leq k1leqileqk, are distinct, with the possible exception that we may have yk=y0=0y_k = y_0 = 0yk=y0=0. We demonstrate the sequenceability of subsets of size kkk of mathbbZnsetminus0\mathbb{Z}_n \setminus \{ 0 \}mathbbZnsetminus0 when n=mtn = mtn=mt in many cases, including when mmm is either prime or has all prime factors larger than k!/2k! /2k!/2 for kleq11k \leq 11kleq11 and tleq5t \leq 5tleq5 and for k=12k=12k=12 and tleq4t \leq 4tleq4. We obtain similar, but partial, results for 13leqkleq1513 \leq k \leq 1513leqkleq15. This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not contain 000 then it is sequenceable.