The merit factor of binary arrays derived from the quadratic character

We calculate the asymptotic merit factor, under all cyclic rotations of rows and columns, of two families of binary two-dimensional arrays derived from the quadratic character. The arrays in these families have size p x q, where p and q are not necessarily distinct odd primes, and can be considered as two-dimensional generalisations of a Legendre sequence. The asymptotic values of the merit factor of the two families are generally different, although the maximum asymptotic merit factor, taken over all cyclic rotations of rows and columns, equals 36/13 for both families. These are the first non-trivial theoretical results for the asymptotic merit factor of families of truly two-dimensional binary arrays.


Introduction
We consider an array of size n × m to be an infinite matrix A = (a ij ) of realvalued elements satisfying a ij = 0 unless 0 ≤ i < n and 0 ≤ j < m.
The array is called binary if a ij takes values only in {1, −1} for all i, j satisfying 0 ≤ i < n and 0 ≤ j < m, and is called ternary if a ij takes values only in {0, 1, −1}. Given integers u and v, the aperiodic autocorrelation of A = (a ij ) at displacement (u, v) is defined to be C A (u, v) := i,j a ij a i+u,j+v .
We refer to an array of size n × 1 as a sequence of length n, abbreviating the array (a i0 ) to (a i ) and the aperiodic autocorrelation C A (u, 0) to C A (u).
Binary arrays with small out-of-phase aperiodic autocorrelation have a wide range of applications in digital communications and storage systems, including radar [1] and steganography [23]. Ideally, we would like to find a binary array A of size n × m satisfying (1) |C A (u, v)| ≤ 1 for all (u, v) = (0, 0), in which case, A is called a Barker array [1]. However, it was recently shown by Davis, Jedwab, and Smith [6] that a (truly two-dimensional) Barker array must have size 2 × 2. (Barker sequences of length n, namely n × 1 Barker arrays, are known for n ∈ {2, 3, 4, 5, 7, 11, 13}, and any other Barker sequence must have even length [22] greater than 10 29 [18]. Since the Barker array criterion (1) is too restrictive for array dimensions exceeding 2 × 2, it is natural to define a measure for the collective smallness of the aperiodic autocorrelation values of a binary array. One such measure is the merit factor, which is defined for a binary array A = (a ij ) of size n × m with nm > 1 to be Let F n,m denote the maximum value of F (A) taken over all 2 nm binary arrays A of size n × m, and abbreviate F n,1 to F n . We note that the mean of 1/F (A), taken over all binary sequences A of length n, equals 1 − 1/n [19]. The argument of [19] easily generalises to two dimensions: the mean of 1/F (A), taken over all 2 nm binary arrays A of size n × m, equals 1 − 1/(nm). It follows that F n,m ≥ nm/(nm − 1), which asymptotically equals 1 and provides a first benchmark result.
A number of theoretical and computational results on F n are known (see [11] for a survey). One line of research is to calculate F n for small values of n. At present, F n has been calculated for all n ≤ 60 (see [11,Fig. 1], for example). The largest values of F n currently known are F 13 = 169 12 14.08 and F 11 = 121 10 , which are attained by Barker sequences. The computational analysis of F n quickly becomes infeasible as n grows. Another line of research is therefore to construct particular infinite families of binary sequences of increasing length and to calculate their asymptotic merit factor.
The only non-trivial infinite families of binary sequences for which the asymptotic value of the merit factor is known are Rudin-Shapiro sequences [17], Legendre sequences [9], and m-sequences [14], together with some generalisations of these families [10], [15], [20], [12], [13]. The largest proven asymptotic merit factor of a binary sequence family is 6, which is attained by cyclically rotated Legendre sequences (see Theorem 2.1). There is also considerable numerical evidence, though currently no proof, that an asymptotic merit factor greater than 6.34 can be achieved for a family of binary sequences related to Legendre sequences [4].
Much less is known about the value of F n,m for n, m > 1. Eggers [7,Tab. 4.2] computed F n,m for nm ≤ 21 and found lower bounds on F n,m for nm ≤ 121. Although the data supplied in [7] are very limited, it is apparent that F n,m tends to be smaller than F nm . The largest value of F n,m for n, m > 1 reported in [7] equals F 4,4 = 16 3 5.33. However, an elementary construction technique gives binary arrays with larger merit factor. Given two sequences A = (a i ) and B = (b j ) of length n and m, respectively, we follow [5] in defining the product array A × B := (a i b j ). A straightforward calculation shows that from which we deduce Let A and B be Barker sequences of length 13 and 11, respectively. It follows from (3) that F (A × A) 6.80, F (A × B) 6.27, and F (B × B) 5.81. Another consequence of (3) is Currently, no theoretical results on the asymptotic merit factor of families of binary truly two-dimensional arrays are known. Bömer and Antweiler [2] analysed the merit factor of several binary array families numerically. Among the investigated families, two types of array families related to the quadratic character appeared to have largest merit factor. The arrays in the first family were proposed by Calabro and Wolf [5] and have size p × q, and the arrays in the second family were proposed by Bömer, Antweiler, and Schotten [3] and have size p × p, where p and q are (not necessarily distinct) odd primes. Both families can be considered as twodimensional generalisations of Legendre sequences. The authors of [2] successively applied three operations, namely rotations of rows and columns, stairlike rotations of rows and columns, and proper decimations, and computed the maximum value of the merit factor for arrays of small sizes taken from these families. They then remarked [2, p. 8] that ". . . , for large arrays, the ACF [aperiodic autocorrelation function] merit factors of both classes [the above mentioned array families of square size] appear to tend to 3.", and asked for a theoretical explanation of this observation.
In this paper we study the merit factor of two families of binary arrays. The arrays in the first family, called Legendre arrays, have size p × q, where p and q are (not necessarily distinct) odd primes, and contain as a special case the arrays proposed by Calabro and Wolf [5]. The arrays in the second family, called quadraticresidue arrays, have size p × p, where p is an odd prime, and contain as a special case the arrays proposed by Bömer, Antweiler, and Schotten [3]. We calculate, under certain conditions on the growth rate of p relative to q, the asymptotic merit factor at all rotations of rows and columns for both array families. In particular, we show that for both families the asymptotic merit factor equals 36 13 2.77 for an optimal rotation of rows and columns. Although we only maximise the merit factor with respect to the first operation considered in [2], namely rotations of rows and columns, this result does not support the conclusion of [3] quoted above. For all other (non-optimal) rotations of rows and columns, the asymptotic merit factor of quadratic-residue arrays is larger than that of square Legendre arrays.

Two Families of Binary Arrays
Given an odd prime p and a positive integer m, let GF(p m ) be the finite field containing p m elements. Whenever convenient, we treat integers after reduction modulo p as elements in GF(p). The quadratic character of GF(p m ) is the function for a = 0 −1 for a not a square in GF(p m ) +1 otherwise.
If m = 1, then (a | p) := χ(a) is the Legendre symbol. A Legendre sequence L = ( i ) of prime length p > 2 is defined by If the initial element in a Legendre sequence is changed to zero, so that then we call L a ternary Legendre sequence.
In what follows, we present two families of binary arrays, which can be considered as two-dimensional generalisations of Legendre sequences. Let p and q be two (not necessarily distinct) odd primes, and let V p,q be the set of ternary arrays V = (v ij ) of size p × q satisfying |v ij | = 1 for (i = 0 and 0 ≤ j < q) or (j = 0 and 0 ≤ i < p) 0 otherwise.
Then V p,q contains 2 p+q−1 arrays, each having p + q − 1 nonzero elements. Given ternary Legendre sequences L and K of length p and q, respectively, we define a Legendre array X of size p × q to be a binary array of size p × q that can be written as For example, the array X = (x ij ) of size p × q, given by for i = 0 and 1 ≤ j < q (i | p)(j | q) for 1 ≤ i < p and 1 ≤ j < q, is a Legendre array of size p × q. This particular array was originally defined by Calabro and Wolf [5], and its merit factor properties were investigated numerically in [2]. In the original paper [5] such an array was called a "quadratic-residue array". We use the term Legendre array to distinguish it from our second family of binary arrays.
Let p be an odd prime, let χ be the quadratic character of GF(p 2 ), and let {α, α } be a basis for GF(p 2 ) over GF(p). Following Bömer, Antweiler, and Schotten [3], we define a quadratic-residue array Y = (y ij ) of size p × p to be a binary array of size p × p satisfying The class of quadratic-residue arrays (y ij ) satisfying y 00 = +1 was defined by Bömer, Antweiler, and Schotten [3], and its merit factor properties were investigated numerically in [2]. In our analysis it will be convenient to change the leading element in a quadratic-residue array to zero. Accordingly, we define the ternary quadraticresidue array of size p × p to be the ternary array Z = (z ij ) of size p × p given by (5) z ij := χ(iα + jα ) for 0 ≤ i, j < p.
Next we define an operation acting on an array to produce a new array of the same size. Given an array A = (a ij ) of size n × m and real numbers s and t, the rotation A s,t is the array B = (b ij ) of size n × m given by (6) b ij = a (i+ ns ) mod n, (j+ mt ) mod m for 0 ≤ i < n and 0 ≤ j < m.
If A is a sequence of length n, we abbreviate A s,0 to A s . The asymptotic merit factor of a Legendre sequence was calculated for all rotations by Høholdt and Jensen [9].
Theorem 2.1 (Høholdt and Jensen [9]). Let L be the Legendre sequence of prime length p > 2, and let s be a real number satisfying − 1 The constraint − 1 2 < s ≤ 1 2 in Theorem 2.1 is for notational convenience only since by definition A s is the same as A s+1 for every sequence A and all real s. The maximum asymptotic merit factor of a rotated Legendre sequence L s is 6, which occurs for s = ± 1 4 .

Calculation of the Merit Factor of an Array
Given a positive integer n, let ζ n := e √ −1 π/n be a primitive (2n)th root of unity. Let A = (a ij ) be an array of size n × m. The generating function of A is defined to be the power series If A is a sequence of length n, we write A(x) for A(x, y). The next lemma shows how the merit factor of A can be computed from the values A(ζ i n , ζ j m ). This approach generalises to two dimensions the method of Høholdt and Jensen [9] to compute the asymptotic merit factor of a sequence.
Proof. Straightforward manipulation shows that and therefore, Using this identity, an elementary calculation gives 1 4nm as required.

The Merit Factor of Legendre Arrays
In this section we compute the asymptotic merit factor of Legendre arrays for all rotations, subject to certain conditions on the growth rate of the dimensions. We first record a result on the aperiodic autocorrelation of rotated ternary Legendre sequences, which arises as an immediate corollary of [20,Thm. 3].
Proposition 4.1. Let L be the ternary Legendre sequence of prime length p > 2, and let s be a real number satisfying − 1 2 < s ≤ 1 2 . Then, as p → ∞, We note that, as explained after [20, Thm. 3], we can recover Theorem 2.1 from Proposition 4.1. We also need the following bound for the magnitude of a polynomial over C at a (2d)th root of unity, in terms of its magnitudes at dth roots of unity.
Proof. The bound is trivial in the case that j is even. We may therefore take j to be odd, writing j = 2 + d for some integer so that ζ j d = −ζ 2 d . It is then sufficient to bound |A(−ζ 2 d )|. Now by Lagrange interpolation we have and so, since d is odd, The result follows from the inequality The next theorem gives, under certain conditions on the growth rate of p relative to q, the asymptotic merit factor of all 2 p+q−1 Legendre arrays of size p × q for all rotations. Let X be a Legendre array of size p × q, and let s and t be real numbers satisfying − 1 2 < s, t ≤ 1 2 . Then Proof. Let L and K be the ternary Legendre sequences of length p and q, respectively, and write T : The condition (8) implies that p and q grow without bound as N → ∞. We can therefore apply Proposition 4.1 to give We claim that (12) ∆(N ) → 0 as N → ∞.
Apply this bound to (13) to obtain Given a ternary Legendre sequence A of length d, it is well known (see [21, p. 182], for example) that |A(ζ 2k d )| ≤ d 1/2 for each integer k. It is easily verified that this implies |A r (ζ 2k d )| ≤ d 1/2 for each integer k and all real r. Therefore, since T s,t (x, y) = L s (x)K t (y), Lemma 4.2 gives T s,t (ζ i p , ζ j q ) ≤ 4(pq) 1/2 log(p + q) for all integers i and j. Substitute into (14) to give (15)
There is no loss of generality in Theorem 4.3 from the restriction − 1 2 < s, t ≤ 1 2 since A s,t is the same as A s+1,t and A s,t+1 for every array A and all real s and t. We note that the condition (8) can be relaxed for particular Legendre arrays. For example, let L and K be the Legendre sequences of length p and q, respectively. Then X = L × K is a Legendre array. From (3) and Theorem 2.1 we conclude that (9) holds under the relaxed condition p → ∞ and q → ∞ as N → ∞.

The Merit Factor of Quadratic-Residue Arrays
In this section our goal is to calculate the asymptotic merit factor of a quadraticresidue array of size p × p at all rotations. We shall assume throughout this section that p is an odd prime. Write the pth roots of unity as j := e √ −1 2πj/p for integer j.
Then, since p is odd, we have Therefore, given an array A of size p × p, Lemma 3.1 asserts that Our objective is to find an asymptotic expression for the sum on the right-hand side of the identity (16), where A is a rotated ternary quadratic-residue array. Since a ternary quadratic-residue array and a quadratic-residue array differ in only one element, this will be sufficient to compute the asymptotic merit factor of a rotated quadratic-residue array. Before we analyse the sum in (16), we shall need several technical results, which we state in the next subsection. Such a basis is guaranteed to exist [16, p. 58]. Given integers i, j, k, , we then have by (17) and (18) Tr((iα + jα )(kβ + β )) = ik + j , and therefore by putting a := iα + jα . The above sum is called a Gaussian sum, and it is well known that (see [16, pp. 199-201], for example). This proves the lemma.
Our next lemma bounds a certain character sum and evaluates it in special cases.

Asymptotic Merit Factor Calculation.
We are now in a position to analyse asymptotic behaviour of the sum on the right-hand side of the identity (16), where A is a rotated ternary quadratic-residue array. We split the analysis into the following three lemmas.
Lemma 5.4. Let Z be a ternary quadratic-residue array of size p × p, and let s and t be real numbers satisfying − 1 2 < s, t ≤ 1 2 . Then, as p → ∞, Proof. Lemma 5.1 implies Then, using the easily verified identity as required.
Lemma 5.5. Let Z be a ternary quadratic-residue array of size p × p, and let s and t be real numbers satisfying − 1 2 < s, t ≤ 1 2 . Then, as p → ∞, Proof. Given an array A, let A T denote the transpose of A. Since (Z s,t ) T = (Z T ) t,s and Z T is again a ternary quadratic-residue array, it is sufficient to prove the first statement in the lemma.
Put x := aβ + jβ and note that aβ + jβ ranges over GF(p 2 ) as a and j range from 0 to p − 1. By the definition (19) of Ω(κ, λ, µ) we therefore obtain Let the set I be as defined in (20), and write Using (22), the sum A can be bounded as Application of (21) and (25) to evaluate Ω(kβ, kβ, 0) and Γ(k, k, 0) then gives We wish to apply the identity for integer j satisfying |j| ≤ p (see [15, p. 621], for example). Since 1 2 < s ≤ 1 2 , we have |2S − 1| ≤ p for all sufficiently large p, which allows us to apply (33) to (32) to obtain, for all sufficiently large p, By the definition of S, we have S = ps + O(1) as p → ∞, and therefore, The proof is completed by substituting (31) and (34) in (30).
Lemma 5.6. Let Z be a ternary quadratic-residue array of size p × p, and let s and t be real numbers satisfying − 1 2 < s, t ≤ 1 2 . Then, as p → ∞, Proof. The idea of the proof is similar to that of the proof of Lemma 5.5. The main difference is that we now have to apply interpolation of Z s,t (x, y) in both indeterminates.
Let i and j be integer. By the interpolation formula (28) we have Apply the interpolation formula (28) again to obtain Set S := ps and T := pt . Then by (27) and Lemma 5.1 we get where χ is the quadratic character of GF(p 2 ) and {β, β } is some basis for GF(p 2 ) over GF(p). Since we also have Use (29), relabel the summation indices, and use the definition (19) of Ω(κ, λ, µ) to give 0≤i,j<p by (26), giving We use symmetry of Ω(κ, λ, µ) and Γ(k, , m) with respect to interchanging their arguments to partition the sum B further as Application of (21) and (25) to evaluate Ω(κ, κ, 0) and Γ(k, k, 0) then gives Using Lemmas 5.4, 5.5, and 5.6 to determine the asymptotic behavior of the sums on the right-hand side, we obtain  The theorem follows from (42) by noting that C Ys,t (0, 0) = p 2 .

Concluding Remarks
We have computed the asymptotic value of the merit factor of Legendre arrays (under certain conditions on the growth rate of their dimensions) and of quadraticresidue arrays, for all rotations of rows and columns. The asymptotic merit factor of rotated Legendre arrays and rotated quadratic-residue arrays is shown in Figures 1 and 2, respectively. The maximum asymptotic merit factor, taken over all rotations, equals 36 13 2.77 for both array families, which occurs at the rotations (s, t), where s, t ∈ { 1 4 , 3 4 }. However, at all other rotations, the asymptotic merit factor of quadratic-residue arrays is larger than that of Legendre arrays. On the other hand, an advantage of Legendre arrays is that they are not restricted to be square.
In [4], the authors exhibited a family of sequences A obtained by appending an initial fraction of a rotated Legendre sequence to itself. Based on partial explanations and extensive numerical computations, it was conjectured in [4] that this sequence family has asymptotic merit factor greater than 6.34. Under the assumption that this conjecture is correct, the corresponding square product array A × A has asymptotic merit factor greater than 2.93, by (3). This suggests that the maximum asymptotic value of the merit factor of the two array families considered in this paper, namely 36 13 2.77, can be surpassed by another array family. In closing, we remark that instead of studying the merit factor of two-dimensional arrays of size n×m, Gulliver and Parker [8] studied the (suitably generalised) merit factor of d-dimensional arrays of size 2×2×· · ·×2. In [8], the merit factor of several families of such arrays was calculated. In particular, the largest asymptotic value, as d → ∞, of the merit factor of a family of d-dimensional arrays considered in [8] equals 3.