The Existence of （v, 4, 1） Disjoint Difference Families with v a Prime Power

A （v, k, λ） difference family （（v, k, λ）-DF in short） over an abelian group G of order v, is a collection F=（Bi｜i ∈ I} of k-subsets of G, called base blocks, such that any nonzero element of G can be represented in precisely A ways as a difference of two elements lying in some base blocks in F. A （v, k, λ）-DDF is a difference family with disjoint blocks. In this paper, by using Weil＇s theorem on character sum estimates, it is proved that there exists a （p^n, 4, 1）-DDF, where p = 1 （rood 12） is a prime number and n ≥1.

The existence of (v, 4, 1) disjoint difference families with v a prime power

A (v, k, λ) difference family ((v, k, λ)-DF in short) over an abelian group G of order v, is a collection F = {B _i |i ∈ I} of k-subsets of G, called base blocks, such that any nonzero element of G can be represented in precisely λ ways as a difference of two elements lying in some base blocks in F. A (v, k, λ)-DDF is a difference family with disjoint blocks. In this paper, by using Weil’s theorem on character sum estimates, it is proved that there exists a (p ⁿ, 4, 1)-DDF, where p = 1 (mod 12) is a prime number and n ≥ 1. Supported by the National Natural Science Foundation of China (No.10561002), Guangxi Science Foundation (No.0640062) and Innovation Project of Guangxi Graduate Education.