New Lower Bounds For The Symmetric L₂-discrepancy And Their Application

Uniform designs (Fang and Wang, 1994;Fang et al, 2000) spread their experimental points evenly throughout the design space, but how to measure the uniformity of a given design? Various discrepancies in quasi-Monte Carlo methods have been often used as measures of uniformity in the literature, such as the star-discrepancy, the star-L₂-discrepancy (Fang and Wang, 1994), the centered L₂-discrepancy, the symmetric L₂-discrepancy and the wrap-around L₂-discrepancy (Hickernell, 1998a,b). Then a design whose discrepancy value achieves a strict lower bound is a uniform design with respect to this kind of discrepancy. So to find a good lower bound is an very important issue in the theory of uniform designs.The present paper aims at obtaining some new lower bounds for the symmetric L₂-discrepancy on some kind of symmetric factorials and asymmetric factorials. Using these lower bounds not only helps to measure the uniformity of given designs but also helps to construct uniform designs. Meanwhile we also give some illustrative examples of uniform and nearly uniform designs to support our theoretical results.The main results of this paper are given as follows: Theorem 1 d ∈ F(n;2^s) is a U-type design, thenwhere h = sn/(2(n - 1)), and the equality holds when h_kl = h, for any k≠l. Here h_kl is the Hamming distance between the k-th and l-th rows of a design d.Theorem 2 d∈ F(n;p₁ × 2^S2) is a U-type design, thenwhere l₀ = (11/8)^S2, and for 0 ≤ i ≤ s₂, 1 ≤ j ≤ p₁ - 1, h_i = 2ⁱ let g_i be the largest integer contained in n/h_i, and g_ij be the largest integer contained in nj/(p₁h_i),...