Difference systems of sets and a collection of 3-subsets in a finite field of order p

作者：Chisaki Shoko^*; Miyamoto Nobuko

来源：Finite Fields and Their Applications, 2015, 34: 75-94.

DOI：10.1016/j.ffa.2015.01.003

• Difference systems of sets
• Cyclotomic coscts
• Cyclotomic numbers

A difference system of set (DSS) is a collection of t disjoint tau(i)-subsets Q(i), 0 <= i <= t-1, of Z(n) such that every non-identity element of Z(n) appears at least rho times in the multiset {a - b vertical bar a is an element of Q(i), b is an element of Q(j), 0 <= i, j <= t - 1, i not equal j}. A DSS is regular if tau(i) is constant for 0 <= i <= t - 1, and a DSS is perfect if every element of Z(n) is contained exactly rho times in the above multiset. In this paper, we consider a collection of 3-subsets of a finite field of a prime order p = ef + 1 to be a DSS. We present a condition for which the collection forms a regular DSS and give a lower bound on the parameter p using cyclotomic numbers for e = 3,4 and 6. For the same values of e, we also show a condition for which a collection of 3-subsets is a perfect DSS.

• 出版日期2015-7