Constructions and Applications of Perfect Difference Matrices and Perfect Difference Families
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Perfect difference families (PDFs for short) are important both in theoretical and in applications. Perfect difference matrices (PDMs for short) and the equivalent structure had been extensively studied and used to construct perfect difference families, radar array and related codes. The necessary condition for the existence of a PDM(n,m) is m≡ 12 and m≥ n+1. So far, PDM(3,m)s exist for odd 5≤ m≤ 201 with two definite exceptions of m=9,11. In this paper, new recursive constructions on PDM(3,m)s are investigated, and it is proved that there exist PDM(3,m)s for any odd 5≤ m<1000 with two definite exceptions of m=9,11 and 33 possible exceptions. A complete result of (g,{3,4},1)-PDFs with the ratio of block size 4 no less than 1/14 is obtained. As an application, a complete class of perfect strict optical orthogonal codes with weights 3 and 4 is obtained.
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