w-cyclic holey group divisible designs and their application to three-dimensional optical orthogonal codes

Motivated by the application to three-dimensional optical orthogonal codes, we consider the constructions for a w-cyclic holey group divisible design of type (u,wv) with block size k, which is denoted by w-cyclic k-HGDD of type (u,wv). The necessary conditions of such a design, namely u,v≥k, (u−1)(v−1)w≡0(modk−1) and u(u−1)v(v−1)w≡0(modk(k−1)), are shown to be sufficient for k=3. As an application, we give a complete solution of the existence problem of perfect three-dimensional optical orthogonal codes of weight three with both at most one-pulse per spatial plane and at most one-pulse per wavelength plane properties.