Difference Unit Groups in ℤ_n

Abstract

In 2004, Ryoh Fuji-Hara et al. (IEEE Trans. Inf. Theory. 50(10):2408-2420, 2004) proposed an open problem of finding a maximum multiplicative subgroup G in ℤn satisfying two conditions: (1) the sum of any two distinct elements in G is nonzero; (2) any difference from G is still a unit in ℤn. The subgroups satisfying Condition (2) is called difference unit group. Difference unit group is related to difference packing, zero-difference balanced function and partitioned difference family, and thus have many applications in coding and communication. Suppose the canonical factorization of n is ∏ki=1 peii. In this letter, we mainly answer the open problem with the result that the maximum cardinality of such a subgroup G is $\frac{d}{2^m}$, where d = gcd(p1 － 1, p2 － 1, ･･･, pk － 1) and m = ν2(d). Also an explicit construction of such a subgroup is introduced.