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In this paper we investigate combinatorial constructions for $w$-cyclicholely group divisible packings with block size three (briefly by $3$-HGDPs).For any positive integers $u,v,w$ with $u\equiv0,1~(\bmod~3)$, the exact numberof base blocks of a maximum $w$-cyclic $3$-HGDP of type $(u,w^v)$ isdetermined. This result is used to determine the exact number of codewords in amaximum three-dimensional $(u\times v\times w,3,1)$ optical orthogonal codewith at most one optical pulse per spatial plane and per wavelength plane.

Maximum $w$-cyclic holely group divisible packings with block size three and applications to optical orthogonal codes