Further progress on difference families with block size 4 or 5. (English) Zbl 1198.94186
This paper under review concerns the existence of \((pq,4,1)\) and \((pq,5,1)\) difference families, for primes \(p\) and \(q\) congruent to 7 modulo 12, and congruent to 11 modulo 20, respectively.
In both cases, it is first shown that under certain conditions, the existence of good \((p,n,1)\) and \((q,n,1)\) (\(n=4\) or \(5\)) optical orthogonal codes guarantees the existence of \((pq,n,1)\) difference family. Then with computer’s help, the conditions are verified to hold for all primes \(p\) and \(q\) not exceeding 1000. Moreover, it is indicated that the examples obtained, using the constructions in the theorems, all have multipliers of order \(n\) fixing a number of base blocks.
In both cases, it is first shown that under certain conditions, the existence of good \((p,n,1)\) and \((q,n,1)\) (\(n=4\) or \(5\)) optical orthogonal codes guarantees the existence of \((pq,n,1)\) difference family. Then with computer’s help, the conditions are verified to hold for all primes \(p\) and \(q\) not exceeding 1000. Moreover, it is indicated that the examples obtained, using the constructions in the theorems, all have multipliers of order \(n\) fixing a number of base blocks.
Reviewer: Wen-Fong Ke (Tainan)
MSC:
| 94B25 | Combinatorial codes |
| 05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |
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