The existence of (v, 4, λ) disjoint difference families

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 each 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 disjoint (v, k, λ)-DF is a difference family with disjoint blocks. In this paper, it is proved that there exists a (v, 4, 1)-DDF for each prime power v ≡ 1 (mod 12) and v ≥ 13. It is also proved that there exists a (v, 4, 2)-DDF for each prime power v ≡ 1 (mod 6) and v ≥ 7. ∗ Research supported in part by NSFC(Grant No. 10561002), Guangxi Science Foundation (0640062) and Innovation Project of Guangxi Graduate Education. D. Wu is also with Keylab of Information Coding and Transmission, Southwest Jiaotong University, Chengdu, 610031, China. 226 D. WU, J. YANG, S. CHEN AND D. LI