Beating the Probabilistic Lower Bound on q-Perfect Hashing

In this paper, we investigate the achievable rate of perfect hash code. For an integer $$q\ge 2$$ , a q-hash code C is block over $$[q]:=\{1,\ldots ,q\}$$ length n in which every subset $$\{{\textbf{c}}_1,{\textbf{c}}_2,\dots ,{\textbf{c}}_q\}$$ q elements separated, i.e., there exists $$i\in [n]$$ such that $$\{\textrm{proj}_i({\textbf{c}}_1),\dots ,\textrm{proj}_i({\textbf{c}}_q)\}=[q]$$ where $$\textrm{proj}_i({\textbf{c}}_j)$$ denotes ith position $${\textbf{c}}_j$$ . Finding maximum size M(n, q) codes n, for given and fundamental problem combinatorics, information theory, computer science. are interested asymptotic behavior problem. Precisely speaking, will focus on quantity $$R_q:=\limsup _{n\rightarrow \infty }\frac{\log _2 M(n,q)}{n}$$ A well-known probabilistic argument shows existence lower bound $$R_q$$ namely $$R_q\ge \frac{1}{q-1}\log _2\left( \frac{1}{1-q!/q^q}\right) $$ (Fredman Komlós, SIAM J Algebr Discrete Methods 5(1):61–68, 1984; Körner, 7(4):560–570, 1986). This still best-known till now except case $$q=3$$ (Körner Marton, Eur Comb 9(6):523–530, 1988). The improved $$R_3$$ was discovered 1988 has been no progress more than 30 years. paper show can be from 4 to 15 all odd integers between 17 25, sufficiently large q.