Difference-Spectrum Completion of Multi-Weight Optical Orthogonal Codes

Variable-weight (also called multi-weight) optical orthogonal codes (VW-OOCs) with maximum correlation one admit a cyclic difference-packing formulation. We characterize the exact difference spectrum of an existing Chinese-remainder graph construction: each mixed-coordinate difference determines a unique candidate base pair and graph index. A pairwise balanced design covers the mixed region exactly, enabling independent completion on the two coordinate axes. For weights {3,4}, punctured finite geometries give families that are optimal for their prescribed profiles. For every prime p≡1(mod12), the length-12p family has (n3,n4)=((p+1)/2,3(p-1)/4), profile (2(p+1),3(p-1))/(5p-1), and deficiency 5. For every prime p≡1(mod30), the length-15p family has (n3,n4)=((p-1)/2,p), profile (p-1,2p)/(3p-1), and deficiency 2. Both deficiencies are the minima permitted by congruence. This study's contribution is the exact-spectrum analysis and its uniform completion mechanism; the lengths and weight set are known, and the second completed parameter tuple follows by filling a previously known regular packing. Explicit examples are verified by exact enumeration.