Motivated by the application to three-dimensional optical orthogonal codes, we consider the constructions for a ww-cyclic holey group divisible design of type (u,wv)(u,wv) with block size kk, which is denoted by ww-cyclic kk-HGDD of type (u,wv)(u,wv). The necessary conditions of such a design, namely u,v≥ku,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=3k=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.
['Suhui Dai', 'Yanxun Chang', 'Lidong Wang']
w-cyclic holey group divisible designs and their application to three-dimensional optical orthogonal codes