Haiping Zhou

A ${t-(v,k,\lambda) }$ {\it packing} is a pair $({X, \cal B})$ where $X$ is a $v$-set, and $\cal B$ is a collection of $k$-subset (called blocks) of $X$, such that any $t$-subset of $X$ occurs in at most $\lambda$ blocks. A $t-(v,k,\lambda) $ packing is {\it cyclic} if its automorphism group contains a cycle of length $v$. Furthermore, if under the action of this automorphism, each orbit is of length $v$ (without any short ones), it is called a strictly cyclic packing. A 2-design is said to be {\it super-simple} if the intersection of any two blocks has at most two elements. A 2-design is called \emph{super-simple cyclic} if it is both cyclic and super-simple. In the sequel, we will use the notation $(v,k,\lambda)$-SCP to denote a super-simple strictly cyclic $(v,k,\lambda)$-packing with maximum number of blocks. When the number of blocks is $\frac{\lambda v(v-1)}{k(k-1)}$, the packing is a super-simple strictly cyclic $(v,k,\lambda)$-BIBD. The study of SCP is motivated by an application in optical orthogonal codes(OOC). Chu has proved that any ${t-(v,k,1)}$ strictly cyclic packing is equivalent to an $(v,k,t-1)$-OOC. In this paper, direct and recursive constructions for $(v,k,\lambda)$-SCP are developed. We solve the existence of a $(v,3,\lambda)$-SCP completely for each $\lambda\in\{2,3,4\}$. We also get some results on $(v,k,\lambda)$-SCPs for $k\ge 4$ and $\lambda\ge 2$.