Bulletin of the
Korean Mathematical Society BKMS

Zero-difference balanced (ZDB) functions can be applied to many areas like optimal constant composition codes, optimal frequency hopping sequences etc. Moreover, it has been shown that the image set of some ZDB functions is a regular partial difference set, and hence provides strongly regular graphs. Besides, perfect nonlinear functions are zero-difference balanced functions. However, the converse is not true in general. In this paper, we use the decomposition of cyclotomic polynomials into irreducible factors over $\mathbb F_p$, where $p$ is an odd prime to generalize some recent results on ZDB functions. Also we extend a result introduced by Claude et al. [3] regarding zero-difference-$p$-balanced functions over $\mathbb F_{p^n}$. Eventually, we use these results to construct some optimal constant composition codes.

Keywords: Zero-difference balanced (ZDB) function, cyclotomic polynomials, cyclotomic coset, constant composition code

MSC numbers: 06E30, 05B10, 94B25