Difference Systems of Sets with Size 2

Abstract

Difference systems of sets (DSS) are combinatorial structures introduced by Levenshtein in (Probl Peredachi Inform 7(3):215–222, 1971), which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we consider a collection of pairs in a finite field of a prime order $p=ef+1$ to be a regular DSS with parameters $(p,2,f,\rho )$. We give a lower bound on the parameter $\rho$ using cyclotomic numbers for $e=3$ and 4. In addition, we present a condition for which the collection of pairs forms an optimal DSS for $e=4$.

Appendices

Appendix A

We define the cyclotomic matrix of order e, $(A_{ij})$ : $0 \le i,j \le e-1$, to be the matrix whose ij-th entry is the integer $(i,j)_e$.

Let $p=3f +1$ be an odd prime, where $4p = x^2 + 27 y^2$ with $x \equiv 1$ (mod 3). Then the cyclotomic numbers of order 3 are given by Table 5 and

$$\begin{aligned} \left\{ \begin{array}{lll} 9A_3 &{}=&{} p -8 +x, \\ 18B_3 &{}=&{} 2p -4 -x -9y, \\ 18C_3 &{}=&{} 2p -4-x +9y, \\ 9D_3 &{}=&{} p+1+x. \end{array} \right. \end{aligned}$$

Table 5 cyclotomic matrix of order 3

Appendix B

Let $p = 4f+1$ be an odd prime satisfying $p = x^2 + 4y^2$ and $x \equiv 1$ (mod 4). When f is odd, the cyclotomic numbers of order 4 are given by Table 6 and

$$\begin{aligned} \left\{ \begin{array}{lll} 16A_4 &{}=&{} p -7 +2x, \\ 16B_4 &{}=&{} p +1+2x-8y, \\ 16C_4 &{}=&{} p+1-6x, \\ 16D_4 &{}=&{} p+1+2x+8y, \\ 16E_4 &{}=&{} p-3-2x. \end{array}\right. \end{aligned}$$

Table 6 cyclotomic matrix of order 4 when f is odd