Some progress on \((v,4,1)\) difference families and optical orthogonal codes
The authors consider collections \({\mathcal C}\) of \(k\)-subsets of a \(v\)-set such that any two sets intersect in at most one point. In the special case \(k=4\), such packings are the same objects as constant-weight-codes with weight \(4\) and minimum distance at least \(6\). If the code associated with the packing is cyclic, it is called an optical orthogonal code (OOC). Given \(v\) and \(k\), there is a trivial upper bound for the number of codewords in an OOC. If this bound is reached, the code is called optimal. Constructions of OOC's use modifications of the well-known difference techniques in design theory (cyclic difference families for designs with blocksize \(4\) are, for instance, special types of OOC's). Using difference methods, the authors construct new families of OOC's of maximum size where \(v\) is prime and satisfies some number theoretic conditions. These new examples are used to construct optimal OOC's of length \(5v\) (if \(v\equiv 5\mod 12\)) and \(7v\) (if \(v\equiv 7\mod 12\)). The latter examples are actually difference families. Moreover, it is shown that cyclic difference families with blocksize \(4\) exist for all \(v\leq 601\), except \(v=25\). Optimal OOC's exist for all \(v\leq 408\), except \(v=25\).
scientific article; zbMATH DE number 2081940