General Lower Bounds on Maximal Determinants of Binary Matrices

Abstract

Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n n/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = n − h, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. For example,By a recent result of Livinskyi, d 2 /h 1/2 → 0 as n → ∞, so the second bound is close to (πe/2) −d/2 for large n. Previous lower bounds tended to zero as n → ∞ with d fixed, except in the cases d ∈ {0, 1}. For d ≥ 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d ≤ 3, so the first bound above shows that R(n) is bounded below by a positive constant (πe/2) −3/2 > 0.1133.

A Survey of the Hadamard Maximal Determinant Problem

“…Even when using the optimal Hadamard matrices for this method (those with maximal excess), the ratio of the determinant obtained to the bound of Corollary 10 tends to zero as n tends to infinity. A remarkable generalisation of this result was obtained by Brent, Osborn and Smith [9], in which multiple rows and columns are added to a Hadamard matrix. Columns are chosen uniformly at random, while the rows added are chosen deterministically.…”

“…There are several other constructions in the literature for matrices of order n ≡ 2 mod 4 with large determinant. Brent and Osborn [8] consider submatrices of order n − 2 of a Hadamard matrix of order n. Brent, Osborne and Smith [9] add two rows and columns to a Hadamard matrix. This work is discussed further in Section 7.1.…”

“…Theorem 29 (Theorem 3.6, [9]). If 0 d 3, and h is the order of a Hadamard matrix then there exists a matrix M of order n = h + d such that…”

Discrete analogues of Macdonald–Mehta integrals

“…Motivated by work in [8] concerning the Hadamard maximal determinant problem [16], the recent papers [6,7] considered various binomial multi-sum identities of which the following two results (the latter being conjectural in [6]) are representative:…”

Evaluation of binomial double sums involving absolute values

“…Motivated by work in [6] concerning the Hadamard maximal determinant problem [9], Brent and Osborn [5] proved the double sum evaluation It should be noted that the difficulty in evaluating this sum lies in the appearance of the absolute value. Without the absolute value, the summation could be carried out straightforwardly by means of the binomial theorem.…”