G-invariant packings approach optimal sizes for optical codes
A $G$-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes
Information Theory
Summary
The authors studied how to arrange certain groups of points so that their patterns look the same when you shuffle points according to a group’s rules. They found conditions that let these arrangements be almost as large as theoretically possible when the number of points gets very big. This helps create special codes used in optical communication that are very efficient and have predictable patterns. Their work also applies to multi-dimensional versions and cyclic variations of these codes.
What this means in practice
- •For optical network engineers: Design highly efficient optical orthogonal codes with near-optimal size to improve communication capacity and reliability.
- •For coding theory developers: Create multidimensional and cyclic constant-weight codes meeting known size bounds for use in advanced error correction and signal design.
Authors
Yeow Meng Chee
Abstract
Let $k>t\ge 2$ be fixed integers, and let $G$ be a permutation group on a set of $v$ points. A $G$-invariant packing is free if every block has $|G|$ distinct images under $G$. We give two conditions on $G$ under which, as $v\to\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\binom{v}{t}/\binom{k}{t}$ blocks. This is a $G$-invariant form of the Erdős--Hanani theorem on asymptotically optimal packings. For semiregular groups $G$, packings with these properties exist if and only if all but $o(v^t)$ of the $t$-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for $t\ge 3$, and by every cyclic semiregular group for $t=2$. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight $w$ and constant correlation $λ$ with $w\geλ+2$, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.