A theorem in finite projective geometry and some applications to number theory. (English) Zbl 0019.00502

Singer, James

Trans. Am. Math. Soc. 43, 377-385 (1938).

Points in a finite projective plane are given homogeneous coordinates in the Galois field \(\mathrm{GF}(p^n)\). Associating each point with a unique element in \(\mathrm{GF}(p^{3n})\) it is shown that there exists a collineation \(C\) such that if \(A\) and \(B\) denote points a power of \(C\) transforms \(A\) into \(B\). It follows that points can be arranged in a “regular” rectangular array (an array in which each row contains all points, each column contains all points of a line, each row is a cyclic permutation of the first). Subscripts assigned to the points and so arrayed show that if \(m\) is a power of a prime there exists a set of \(m+1\) integers (called a “perfect difference set”) \(d_0, d_1,\ldots, d_n\) such that the \(m^2+m\) differences \(d_i-d_j\) \((i\neq j;\;i,j = 0, 1, \ldots, m)\) are congruent, modulo \(q= m^2+m+1\), to the integers \(1, 2, \ldots, m^2+m\). This leads to a perfect partition \(a_0, a_1, \ldots, a_m\) of \(q\); i. e., each residue class modulo \(q\) is represented uniquely by a circular sum of the \(a\)’s (sum of consecutive \(a\)’s). The number of perfect difference sets corresponding to a given \(q\) is discussed. A partial list of perfect difference sets and perfect partitions is given. The concepts are generalized for \(q=p^{kn} + p^{(k-1)n} + \ldots+ p^n + 1\).

Reviewer: J. L. Dorroh

MSC:

11B75	Other combinatorial number theory
51E20	Combinatorial structures in finite projective spaces
05B10	Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)

Keywords:

finite projective planes; Galois field; perfect difference sets; perfect partitions