Near Zero-diference Balanced Functions And Their Constructions

2008§Ding ([6]) introduced a class of combinatorial functions called zero-diferencebalanced (ZDB) functions in his investigation on constant composition codes. Subsequentresearch showed that zero-diference balanced functions are very useful in constructions ofdiference systemss of sets, partitioned diference families and other combinatorial structure.As a generalization of zero-diference balanced functions, we propose the new notion of anear zero-diference balanced (NZDB) functions.Let (A,+) and (B,+) be two abelian groups of order n and m. Let λ be a non-negativeinteger and f a map from (A,+) to (B,+). For each a∈A\{0}, define Da(x)=f (x+a) f (x)(x∈A). We say that f is an (n, λ; u)-NZDB function, if|{x∈A: Da(x)=0}|=λ or λ+1, a∈A\{0},where, u=|{a∈A\{0}:|{x∈A: Da(x)=0}|=λ}|.In the particular case where u=n1, an (n, λ; u)-NZDB function f is just an (n, λ)-ZDBfunction. In the first chapter, we introduce necessary definitions and the background of theinvestigation. In the second chapter, we reveal an equivalence between NZDB functions andalmost diference families of partition-type. We then take advantage of this equivalence toestablish several constructions of NZDB functions by using theory of cyclotomy and rela-tively diference families of partition type in Chapter3. The main results are presented inTheorem3.8, Theorem3.9, Theorem3.12, Theorem3.20, Theorem3.21.