Beating the probabilistic lower bound on perfect hashing

For an integer q > 2, a perfect q-hash code C is a block code over [q]:= {1,..., q} of length n in which every subset {c1, c2,..., cq} of q elements is separated, i.e., there exists i ∈ [n] such that {proji(c1),..., proji(cq)} = [q], where proji(cj) denotes the ith position of cj. Finding the maximum size M(n, q) of perfect q-hash codes of length n, for given q and n, is a fundamental problem in combinatorics, information theory, and computer science. In this paper, we are interested in asymptotical behavior of this problem. More precisely speaking, we will focus on the quantity Rq := lim supn→∞log2 M(n,q). n A well-known probabilistic argument indicates Rq > q−11 log2 ( 1−q1!/qq ) [10, 12]. This is still the best-known lower bound so far except for the case q = 3 for which Körner and Matron [13] found that the concatenation technique could lead to perfect 3-hash codes that could beat this probabilistic lower bound. This improved lower bound on R3 was discovered in 1988 and there has been no progress of this lower bound on Rq for more than 30 years despite of some work on upper bounds on Rq. In this paper we show that this probabilistic lower bound can be improved for q = 4, 8 and all odd integers between 5 and 25,1 and all sufficiently large q with q (mod 4) 6= 2. Although we are not able to prove that our construction can beat the probabilistic method for all q with q (mod 4) 6= 2, the fact that our construction beat the probabilistic method for both small and large q sheds light on that our new construction might beat the previous lower bound for all q with q (mod 4) 6= 2. Our idea is based on a modified concatenation differing from the concatenation [10] where both the inner and outer codes are separated. In our concatenation, the inner code is not necessarily a perfect q-hash code. This gives a more flexible choice of inner codes and hence we are able to improve the lower bound on Rq.

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