Counting Results for Thin Butson Matrices

A partial Butson matrix is a matrix H ∈ MM×N (Zq) having its rows pairwise orthogonal, where Zq ⊂ C× is the group of q-th roots of unity. We investigate here the counting problem for these matrices in the “thin” regime, where M = 2, 3, . . . is small, and where N → ∞ (subject to the condition N ∈ pN when q = pk > 2). The proofs are inspired from the de Launey-Levin and Richmond-Shallit counting results. Introduction A partial Hadamard matrix is a matrix H ∈ MM×N(±1) having its rows pairwise orthogonal. These matrices are quite interesting objects, appearing in connection with various questions in combinatorics. The motivating examples are the Hadamard matrices H ∈MN(±1), and their M ×N submatrices, with M 6 N . See [9]. A given partial Hadamard matrix H ∈ MM×N(±1) can complete or not into an Hadamard matrix H̃ ∈ MN(±1). It is known since Hall [3] and Verheiden [11] that this automatically happens when K = N −M is small, and more precisely when K 6 7. The structure of such matrices is very simple up to M = 4, where, up to assuming that the first row has 1 entries only, and then permuting the columns, the matrix is: H =  + + + + + + + + + + + + − − − − + + − − + + − − + }{{} a − }{{} b + }{{} b − }{{} a + }{{} b − }{{} a + }{{} a − }{{} b  Here a, b ∈ N are subject to the condition a+ b = N/4. the electronic journal of combinatorics 21(3) (2014), #P3.12 1 At M > 5 no such result is available, and the partial Hadamard matrices give rise to interesting combinatorial structures, related to the Hadamard Conjecture. See Ito [4]. In their breakthrough paper [7], following some previous work in [6], de Launey and Levin proposed a whole new point of view on these matrices, in the asymptotic limit N ∈ 4N, N →∞. Their main result is as follows: Theorem (de Launey-Levin [7]). The probability for a random H ∈MM×N(±1) to be partial Hadamard is PM ' 2(M−1) 2 √ (2πN)( M 2 ) in the N ∈ 4N, N →∞ limit. The proof in [7] uses a random walk interpretation of the partial Hadamard matrices, then the Fourier inversion formula, and then some real analysis methods. Importantly, as pointed out there, this method can be probably used for more general situations. An interesting generalization of the Hadamard matrices are the complex Hadamard matrices H ∈ MN(C) having as entries the roots of unity, introduced by Butson in [2]. The basic example here is the Fourier matrix, FN = (w ) with w = e : FN =  1 1 1 . . . 1 1 w w . . . wN−1 . . . . . . . . . . . . . . . 1 wN−1 w2(N−1) . . . w(N−1) 2  In general, the theory of Butson matrices can be regarded as a “non-standard” branch of discrete Fourier analysis. For a number of results on these matrices, see [10]. We can of course talk about partial Buston matrices: Definition. A partial Butson matrix is a matrix H ∈MM×N(Zq) having its rows pairwise orthogonal, where Zq ⊂ C× is the group of q-roots of unity. Observe that at q = 2 we obtain the partial Hadamard matrices. In general, the interest comes from the Butson matrices H ∈MN(Zq), and from their M×N submatrices. Let us first discuss the case q = 2. At M = 2, up to assuming that the first row has 1 entries only, and then permuting the columns, the matrix must be as follows: H =  1 1 . . . 1 1 1 . . . 1 1 }{{} a1 w }{{} a2 . . . wq/2−1 } {{ } aq/2 w }{{} a1 w } {{ } a2 . . . wq−1 }{{} aq/2  Here w = e and a1, . . . , aq/2 ∈ N are certain multiplicities, summing up to N/2. Thus counting such objects is the same as counting abelian squares, i.e. length N words of type xx′ where x′ is a permutation of x. According now to [8], we have: the electronic journal of combinatorics 21(3) (2014), #P3.12 2 Theorem (cf. Richmond-Shallit [8]). For q = 2 the probability for a randomly chosen H ∈M2×N(Zq) to be partial Butson is P2 ' 2 √( q/2 2πN )q/2 in the N ∈ 2N, N →∞ limit. There are actually several proofs of this result, but the one in [8] is remarkably beautiful: based only on the Stirling formula, and on an old idea of Lagrange. Indeed: P2 = 1 qN ( N N/2 ) ∑ a1+...+aq/2=N/2 ( N/2 a1, . . . , aq/2 )2 The point now is that the sum on the right can be estimated by making a clever use of the Stirling formula, and this gives the above result. See [8]. Summarizing, there are several techniques for dealing with the counting problem for partial Butson matrices. In this paper we will try to use and mix these techniques. Our first result here will be an extension of the Richmond-Shallit count: Theorem A. When q = p is a prime power, the probability for a randomly chosen H ∈M2×N(Zq), with N ∈ pN, N →∞, to be partial Butson is: P2 ' √ p q p q q p