2024/10/30 by Andrea C. Burgess, Burgess, Andrea C., Peter Danziger +3
Engineering · #05B40 #Advanced Manufacturing and Logistics Optimization #Combinatorics (math.CO) #FOS: Mathematics #Manufacturing Process and Optimization #Optimization and Packing Problems
paper · pdf · doi:10.48550/arxiv.2410.22607
openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Given positive integers v, k, t and λ with v ≥ k ≥ t, a packing design PDλ(v,k,t) is a pair (V,B), where V is a v-set and B is a collection of k-subsets of V such that each t-subset of V appears in at most λ elements of B. When λ=1, a PD1(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 PDλ(v,k,t) is called the packing number, denoted PDNλ(v,k,t). In this paper we consider packing designs with k large relative to v. We prove that for a positive integer n, PDNλ(v,k,t) = n whenever nk-(t-1)\binomnλ+1 ≤ λv < (n+1)k-(t-1)\binomn+1λ+1. We also prove that if no point appears in more than three blocks, then the blocks of a PD2(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-\binomn3 ≤ 2v < (n+1)k-\binomn+13. Such directed packing designs yield (k-t)-insertion/deletion codes.