Packing Designs with large block size

TL;DR

This paper investigates packing designs with large block sizes, establishing formulas for their maximum sizes and exploring their applications in coding theory, including directed packing designs and insertion/deletion codes.

Contribution

It provides explicit bounds for the packing number of large block size designs and introduces a method to order blocks to create directed packing designs.

Findings

Exact formulas for packing numbers based on parameters

Construction of directed packing designs with optimal size

Application to insertion/deletion coding schemes

Abstract

Given positive integers $v$, $k$, $t$ and $\lambda$ with $v \geq k \geq t$, a packing design PD$_{\lambda}(v,k,t)$ is a pair $(V,\mathcal{B})$, where $V$ is a $v$-set and $\mathcal{B}$ is a collection of $k$-subsets of $V$ such that each $t$-subset of $V$ appears in at most $\lambda$ elements of $\mathcal{B}$. When $\lambda=1$, a PD$_1(v,k,t)$ is equivalent to a binary code with length $v$, minimum distance $2(k-t+1)$ and constant weight $k$. The maximum size of a PD$_{\lambda}(v,k,t)$ is called the {packing number}, denoted PDN$_{\lambda}(v,k,t)$. In this paper we consider packing designs with $k$ large relative to $v$. We prove that for a positive integer $n$, PDN$_{\lambda}(v,k,t) = n$ whenever $nk-(t-1)\binom{n}{\lambda+1} \leq \lambda v < (n+1)k-(t-1)\binom{n+1}{\lambda+1}$. We also prove that if no point appears in more than three blocks, then the blocks of a PD$_2(v,k,2)$ can be ordered so that no ordered pair occurs more than once. This produces a directed packing design and we show that the corresponding directed packing number is equal to $n$ when $nk-\binom{n}{3} \leq 2v < (n+1)k-\binom{n+1}{3}$. Such directed packing designs yield $(k-t)$-insertion/deletion codes.