The completion of optimal (3,4)(3,4)-packings

A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of quadruples $d(3,4,n)$ denotes the number of blocks in a maximum $3$-$(n,4,1)$ packing design, which is also the maximum number $A(n,4,4)$ of codewords in a code of length $n$, constant weight $4$, and minimum Hamming distance 4. In this paper the undecided 21 packing numbers $A(n,4,4)$ are shown to be equal to Johnson bound $J(n,4,4)$ $( =\lfloor\frac{n}{4}\lfloor\frac{n-1}{3}\lfloor\frac{n-2}{2}\rfloor\rfloor\rfloor)$ where $n=6k+5$, $k\in \{m:\ m$ is odd, $3\leq m\leq 35,\ m\neq 17,21\}\cup \{45,47,75,77,79,159\}$

Optical orthogonal codes obtained from conics on finite projective planes

AbstractOptical orthogonal codes can be applied to fiber optical code division multiple access (CDMA) communications. In this paper, we show that optical orthogonal codes with auto- and cross-correlations at most 2 can be obtained from conics on a finite projective plane. In addition, the obtained codes asymptotically attain the upper bound on the number of codewords when the order q of the base field is large enough