A REVIEW OF THE AVAILABLE CONSTRUCTION METHODS FOR GOLOMB RULERS. Konstantinos Drakakis. (Communicated by Cunsheng Ding)

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1 Advances in Mathematics of Communications Volume 3, No. 3, 2009, doi: /amc A REVIEW OF THE AVAILABLE CONSTRUCTION METHODS FOR GOLOMB RULERS Konstantinos Drakakis UCD CASL University College Dublin Belfield, Dublin 4, Ireland (Communicated by Cunsheng Ding) Abstract. We collect the main construction methods for Golomb rulers available in the literature along with their proofs. In particular, we demonstrate that the Bose-Chowla method yields Golomb rulers that appear as the main diagonal of a special subfamily of Golomb Costas arrays. We also show that Golomb rulers can be composed to yield longer Golomb rulers. 1. Introduction A Golomb ruler can be described as a collection of markings spaced apart at integer multiples of a unit length, so that all distances between pairs of markings are distinct. This is equivalent to saying that, when a Golomb ruler is superimposed to a shifted version of itself (by an integer multiple of the unit length), at most one pair of markings will coincide: in engineering terminology, Golomb rulers have an 1D ideal thumbtack autocorrelation (or auto-ambiguity). Golomb rulers are the same objects as Sidon sets (which historically predate Golomb rulers), namely sets of integers such that all pairwise sums of elements of the set are distinct [21]. This observation was not made immediately by the respective research communities, however, which, in the meantime, had been working separately, producing different bodies of literature, and even duplicate results [7]. Golomb rulers have important applications in engineering, for example in radiofrequency allocation for avoiding third-order interference [1], in generating convolutional self-orthogonal codes [18], in the formation of optimal linear telescope arrays in radio-astronomy [2] etc., as well as an inherent mathematical interest. They owe their name to the studies of Prof. S. Golomb, who demonstrated an important application in graph labeling [13]. Many different construction methods have been proposed, aiming, in particular, to the construction of Golomb rulers as densely populated with markings as possible. A 2D object related to Golomb rulers is the Costas array, namely a square arrangement of dots and blanks such that there is exactly one dot per row and column (i.e. a permutation array), and such that no four dots form a parallelogram and no three dots lying on the same straight line are equidistant. Equivalently, all vectors 2000 Mathematics Subject Classification: Primary: 05B10, 11B50; Secondary: 51E15, 05B20. Key words and phrases: Golomb ruler, Costas array, Golomb construction, Erdös-Turan construction, Rusza-Lindström construction, Bose-Chowla construction, Singer construction. The author is affiliated with UCD CASL and the School of Electronic, Electrical & Mechanical Engineering, University College Dublin (UCD), and also with the Claude Shannon Institute for Discrete Mathematics, Coding, and Cryptography. 235 c 2009 AIMS-SDU

2 236 Konstantinos Drakakis between pairs of dots must be distinct. A Costas array has a 2D ideal thumbtack autocorrelation. In particular, any diagonal of a Costas array is a Golomb ruler. In this paper, we revisit the problem of the efficient construction of Golomb rulers. After presenting the relevant definitions (Section 2), we collect all major known construction methods for Golomb rulers along with their proofs and some examples (Section 3), which are otherwise scattered in the literature and relatively hard to locate; we note, in particular, that one of this methods is equivalent to the Bose-Chowla method, and this equivalence reveals that Bose-Chowla constructed Golomb rulers are obtainable from the main diagonal of a symmetric subfamily of Golomb Costas arrays (Section 5). After discussing briefly optimal Golomb rulers (Section 4), we finally show that Golomb rulers can be (multiplicatively) composed to yield longer Golomb rulers (Section 6). The aim of this paper is twofold: on the one hand, we review the existing construction methods for Golomb rulers, collecting them together along with their proofs, as some of the original papers are not easy to find; on the other hand, we show that Golomb rulers obtained from the Bose-Chowla construction can alternatively be obtained as the main diagonals of a special subfamily of symmetric Golomb Costas arrays. 2. Basics on Golomb rulers and Costas arrays We begin with a formal definition of Golomb rulers; a very comprehensive source of information about them is A. Dimitromanolakis diploma thesis [7]. We use below the shorthand [n] for the set of integers {0, 1,..., n}, along with various obvious variations (such as [n] = {1,...,n} etc.) Definition 1. Let m, n N, and let f : [m 1] [n] be injective with f(0) = 0, f(m 1) = n (whence m n+1); f is a Golomb ruler of length n with m markings if and only if i, j, k, l [m 1], f(i) f(j) = f(k) f(l) i = k j = l. If, in addition, there exists some N N such that i, j, k, l [m 1], f(i) f(j) f(k) f(l) mod N i = k j = l, f will be called a Golomb ruler modulo N, or simply a modular Golomb ruler, whenever N is clear from the context. Remark 1. Geometrically, the Golomb ruler can be interpreted as a sequence of dots (called the markings of the ruler) and blanks, so that all distances between pairs of dots are distinct; the positions of the dots in the sequence (which we will also be calling a Golomb ruler, as there is no danger of confusion) correspond to the range of f. Clearly the Golomb ruler property remains invariant under affine transformations and linear shifts: if f is a Golomb ruler, af( c) + b is also a Golomb ruler, for a Z, b, c Z. This means, in particular, that we can alternatively normalize f so that f : [m] [n + 1] with f(1) = 1. The definition of the Golomb ruler s length above coincides with the usual definition of the length of a (physical) ruler. Note that any modular Golomb ruler is also a Golomb ruler, and that any Golomb ruler is also a Golomb ruler modulo N for a sufficiently large N; the definition is really interesting for small N, in particular N < 2n + 1. Two important questions arise:

3 Construction methods for Golomb rulers What is the maximal m possible for given n? 2. What is the minimal n possible for given m? We will call a Golomb ruler optimally dense if and only if it exhibits the maximal m for a given n and optimally short if and only if it exhibits the minimal n for a given m. Optimally short Golomb rulers are referred to simply as optimal Golomb rulers in the literature, though one might argue that optimally dense Golomb rulers have an equal right to optimality. At any rate, for neither case do we have closed form answers, although several estimates exist: the simplest one is that a Golomb ruler of length n defines at most n possible distances, and, in order to have m points, the ( m 2 (1) ) distances they define will need to be unique. It follows that ( ) m m(m 1) = n m 2n asymptotically. 2 2 This turns out to be a very generous upper bound: improved arguments show that m < n + O(n 1/4 ) [11] and even better that m < n + n 1/4 + 1 [16]. Furthermore, the maximal m for a given n satisfies asymptotically m > n O(n 5/16 ) [11], but it is conjectured to satisfy m > n [7]. Definition 2. Consider a family of Golomb rulers of length n I N, I = ; denote the corresponding number of markings by m(n). The family will be n called (asymptotically) c-nearly optimal if and only if lim = c > 1, and n m(n) n asymptotically optimal if and only if lim n m(n) = 1. Let us now continue with the definition of a Costas function/permutation [5, 6, 8]: Definition 3. Consider a bijection f : [n] [n] ; f is a Costas permutation if and only if: i, j, k such that 1 i, j, i + k, j + k n : f(i + k) f(i) = f(j + k) f(j) i = j or k = 0. A permutation f corresponds to a permutation array A f = [a f i,j ] by setting the elements of the permutation to denote the positions of the (unique) 1 in the corresponding column of the array, counting from top to bottom: a f f(i),i = 1. It is customary to represent the 1s of a permutation array as dots and the 0s as blanks. From now on the terms array and permutation will be used interchangeably, in view of this correspondence. The Costas property is invariant under horizontal and vertical flips, as well as transpositions around the diagonals (and therefore also under rotations of the array by multiples of 90 o, which can be expressed as combinations of the previous two operations), hence a Costas array gives birth to an equivalence class that contains either eight Costas arrays, or four if the array happens to be symmetric. There are two algebraic construction techniques for Costas arrays: the Welch construction [8, 14, 15], which we have no further use for in this work, and the Golomb construction, which we now state without proof, as it is relevant to the discussion below:

4 238 Konstantinos Drakakis Theorem 1 (Golomb construction G 2 (q, a, b)). Let q = p m, where p is a prime and m N, and let a, b be primitive roots of the finite field F(q); the function f : [q 2] [q 2] defined through the equation a i + b f(i) = 1, i [q 2] is a Costas permutation, which corresponds to a symmetric Costas array if and only if either a = b or q = r 2, b = a r ; in the latter case, the resulting symmetric Costas array has r dots on its main diagonal. Proof. See [8, 14, 10]. 3. Construction methods for Golomb rulers Various algebraic methods exist for the construction of Golomb rulers, which appear scattered in the literature. For the benefit of the reader, we collect here the main construction methods used, along with their proofs, giving this section the character of a mini-tutorial (most methods, along with some proofs, appear in [7]). Note that Golomb rulers are not (in a sense about to be specified) well defined, and this has an impact on their construction methods: indeed, for a fixed length n, there exists a maximal number of markings m max (n) a Golomb ruler can have (which lies close to, and perhaps a little bit above, n), and, consequently, whenever m m max (n), a construction method for this m can be found. In particular, for small values of m (relative to m max (n)), many different methods can be found relatively effortlessly (the reader can possibly think of some for oneself). Compare this to the case of Costas arrays, where the number of dots present is regulated by the definition. The classical construction methods of interest tend, as is expected, to produce asymptotically optimal families of Golomb rulers: such families are the most difficult to construct, the most appealing ones mathematically, and often the most useful ones in applications. Most (but not all) constructions follow the same principle: assuming f is to be shown to be a Golomb ruler, we form either of the equations f(x) ± f(y) = a, aiming to show that {x, y} is unique, e.g. by showing that x and y satisfy a polynomial equation of degree 2, or a 2 2 nonsingular system. Let us now review these methods in detail Erdös-Turan construction [11]. Theorem 2. For every odd prime p, the sequence (2) 2pk + (k 2 mod p), k [p 1] forms a Golomb ruler. Proof. We need to demonstrate that the integers x, y that, for a given a N, satisfy (3) 2p(x + y) + (x 2 mod p) + (y 2 mod p) = a can be chosen in at most one way (within the specified range). Dividing throughout by 2p we get a (4) x + y = := A, 2p while taking (3) modulo p we obtain (5) x 2 + y 2 a (x + y) 2 2xy xy 1 2 [(x + y)2 a] : B.

5 Construction methods for Golomb rulers 239 But (4) and (5) imply that x + y and xy are known modulo p, whence they are uniquely determined as roots of the polynomial equation (6) z 2 Az + B 0 mod p, which either has no root or two roots. Therefore, for those a for which roots exist, they are uniquely determined modulo p, and consequently in N due to the assumption that they lie in [p 1]. This completes the proof. The approximate asymptotic length of such a Golomb ruler with p markings is 2p 2, hence the method is 2-nearly optimal. Example 1. Choose p = 13; the resulting sequence is , a Golomb ruler of 13 markings and length Rusza-Lindström construction [16, 19]. Theorem 3. Let p be prime, g a primitive root of F(p), and s relatively prime to p 1. The following sequence (7) (psk + (p 1)g k ) mod p(p 1), k [p 2] forms a Golomb ruler. Proof. This is a very clever construction based on the interplay between two different modulo operators. Consider the equation (8) (psx + (p 1)g x ) mod p(p 1) + (psy + (p 1)g y ) mod p(p 1) = a. Applying modulo p throughout we get (9) g x + g y a mod p, while applying modulo p 1 throughout we get (10) s(x+y) a mod (p 1) x+y s 1 a mod (p 1) g x g y = g s 1a mod p. Taken together, (9) and (10) imply that {g x, g y } is the set of roots of a polynomial equation of degree 2, hence it is determined in at most one way. This completes the proof. These rulers are of length (at most) p(p 1) 1 = p 2 p 1 and have p 1 markings, hence they are asymptotically optimal. This method was first suggested by I. Rusza [19], who studied the special case where s = 1; B. Lindström subsequently generalized it to arbitrary s such that (s, p 1) = 1 [16]. Example 2. Let us choose p = 13, s = 5, g = 2; the resulting sequence is which can be renormalized by a shift to , , a Golomb ruler of 12 markings and length 142.

6 240 Konstantinos Drakakis 3.3. Bose-Chowla construction [3, 4]. Theorem 4. Let q = p n be a power of a prime and g a primitive root in F(q 2 ). Then the q integers in (11) S = {i [q 2 2] : g i g F(q)} have distinct pairwise differences modulo q 2 1. In addition, the set of q(q 1) pairwise differences in S, reduced modulo q 2 1, equals the set of all nonzero integers less than q 2 1 which are not divisible by q+1. Proof. Let x i S, i [4]. It follows that (12) x 1 + x 2 = x 3 + x 4 g x1+x2 = g x3+x4 (g + u 1 )(g + u 2 ) = (g + u 3 )(g + u 4 ), u i F(q), i [4] (u 1 + u 2 u 3 u 4 )g = u 3 u 4 u 1 u 2. Assuming u 1 +u 2 u 3 u 4 0, it follows that g F(q), which is impossible as g is a primitive root of F(q 2 ). Therefore, u 1 +u 2 = u 3 +u 4, whence u 3 u 4 = u 1 u 2 : in other words, the sets {u 1, u 2 } and {u 3, u 4 } satisfy each the same polynomial equation of degree 2, hence they must be identical. This, in turn, implies that the sets x 1, x 2 and x 3, x 4 must be identical. Furthermore, using the same notation, g x1 x2 = g + u 1 ; if g x1 x2 F(q), it g + u 2 would follow that g F(q), a contradiction. But it is easy to see that g i F(q) q + 1 i: indeed, letting i = (q + 1)j, j [q 2], we observe that all these powers are different (as g is a primitive root of F(q 2 ) in the first place), and, ( g i) q 1 = ( ) g j(q 1)(q+1) j = g q2 1 = 1 j = 1, whence g i F(q). Since F(q) has q 1 nonzero elements in total, this proves our claim. It follows that q + 1 cannot divide x 1 x 2 and this completes the proof. These rulers are asymptotically optimal: they have q markings and their length is (at most) q 2 3. Example 3. Let q = 4; the elements of the field of 16 elements can be represented as polynomials in x, whose addition and multiplication take place modulo 2 and a primitive polynomial in x of degree 4, say x 4 + x + 1. We may choose g = x to be our primitive root, in which case consecutive powers of x are given by the following table: x 2 x 2 3 x 3 4 x x 2 + x 6 x 3 + x 2 7 x 3 + x x x 3 + x 10 x 2 + x x 3 + x 2 + x

7 Construction methods for Golomb rulers x 3 + x 2 + x x 3 + x x F(4) is a subfield of F(16), and has g q+1 = x 5 as its primitive root. We then need to find the set of exponents i such that g i g = x i x {0, 1, x 5, x 10 } x i x + {0, 1, x 2 + x, x 2 + x + 1} x i {x, x + 1, x 2, x 2 + 1}, whence i {1, 2, 4, 8}. Therefore the sequence 1, 2, 4, 8, or, equivalently, 0, 1, 3, 7, is a Golomb ruler of four markings and length 7. Remark 2. This method is often referred to as the affine plane construction of Golomb rulers in the literature Singer construction [22]. Singer s result is actually one of the cornerstones of projective geometry. We present it trying to avoid, whenever possible, the use of geometric terminology. Theorem 5. Let q = p n be a power of a prime. Then there exist q + 1 integers {d i : i = 0,..., q} such that the q 2 + q differences d i d j, i, j = 0,...,q, i j, are all distinct; in particular, they coincide with the nonzero integers modulo q 2 + q + 1 when reduced modulo q 2 + q + 1. Proof. Let g be primitive root of F(q 3 ): as such, it satisfies a polynomial equation of degree 3, say (13) g 3 = ag 2 + bg + c, c 0, a, b, c F(q). We may then set (14) g i = x i 1g 2 + x i 2g + x i 3, i [q 3 2], where a direct calculation shows that, for i 0, (15) x i+1 1 = ax i 1 + xi 2, xi+1 2 = bx i 1 + xi 3, xi+1 3 = cx i 1, and (x0 1, x0 2, x0 3 ) = (0, 0, 1). In other words, we can set x i+1 = (x i+1 1, x i+1 2, x i+1 3 ) T = Ax i, where (16) A = a 1 0 b 0 1 A 1 = 0 0 1/c 1 0 a/c. c b/c We define 2 families of sets on the exponents i [q 3 2] of the elements of F(q 3 ) : First, let S k = {i [q 3 2] : i = k+j(q 2 +q+1), j [q 2]}, k [q 2 +q]; in other words, the mutually disjoint S k form the partition resulting from considering two elements of F(q 3 ) equivalent if their ratio belongs in F(q) (note that h = g q2 +q+1 is a primitive root of F(q), since h q 1 = 1 and q 1 is the smallest integer with this property). We will refer to these sets as equivalence classes. Second, consider F(q 3 ) as a 3D vector space over F(q); we can represent its elements as F(q 3 ) = {x = (x 1, x 2, x 3 ) : x 1, x 2, x 3 F(q)}, where x are the expansion coefficients of an element over the basis {1, g, g 2 }, as given by (14); choosing u F(q 3 ), we consider the set T u = {i [q3 2] : x = g i, u x = u T x = u 1 x 1 + u 2 x 2 + u 3 x 3 = 0}. These sets are not mutually disjoint and, what s more, they are

8 242 Konstantinos Drakakis not even distinct. In order to count the number of distinct T u, we determine the number of non-equivalent u by distinguishing cases: if u 1 0, (17) u 1 x 1 + u 2 x 2 + u 3 x 3 = 0 x 1 + u 2 u 1 x 2 + u 3 u 1 x 3 = 0 and there are q 2 possible choices; if u 1 = 0, u 2 0, (18) u 1 x 1 + u 2 x 2 + u 3 x 3 = 0 x 2 + u 3 u 2 x 3 = 0, and there are q possible choices; and if u 1 = u 2 = 0, u 3 0, (19) u 1 x 1 + u 2 x 2 + u 3 x 3 = 0 x 3 = 0, and there is one possible choice; hence there are q 2 + q + 1 distinct sets T u, corresponding to u(k) = g k, k [q 2 + q], which we relabel T k. How many elements does each T k have? Assuming x 1 0, (20) u 1 x 1 + u 2 x 2 + u 3 x 3 = 0 u 1 + x 2 x 1 u 2 + x 3 x 1 u 3 = 0, and there are q(q 1) possible solutions; assuming x 1 = 0, x 2 0, (21) u 1 x 1 + u 2 x 2 + u 3 x 3 = 0 u 2 + x 3 x 2 u 3 = 0, and there are (q 1) possible solutions; and assuming x 1 = x 2 = 0, it follows that x 3 = 0, which we do not allow; hence there are (q 1)(q + 1) solutions, which correspond to q + 1 equivalence classes S d, and therefore to q + 1 indices D = {d i : i [q]}. Note further that any two T k and T l always have a common equivalence class; but if they have two equivalence classes in common, it follows that T k = T l. Fix now u F(q 3 ), and consider the sets C i := (D + i) mod (q 2 + q + 1), i [q 2 + q]; the equality u T x = 0 = u T A 1 Ax = [(A 1 ) T u] T Ax shows that C i = T k for some k iff C i+1 = T l, for some l, while the fact that g is a primitive root, along with the definition of S k, implies that all C i, i [q 2 + q] are different. Retaining the notation above, focus on the sets C di, i [q]: they all contain 0, so they have no other index in common, whence the union q (22) C di \{0} = {d i d j mod (q 2 + q + 1) : i, j [q], i j} i=0 contains q(q + 1) elements, all distinct. Hence these numbers must also be distinct without the application of the modulo operator, and this finishes the proof. These rulers are asymptotically optimal: they have q + 1 markings and their length is at most q 2 + q + 1. Example 4. It is unfortunately impractical to show here all the steps involved in the Singer construction of a Golomb ruler: it is necessary to construct F(q 3 ), which is too big even for moderate values of q. A Singer constructed Golomb ruler for q = 16, hence with q + 1 = 17 markings, is the sequence The ruler s length is 201, and the reader can verify directly that the = 272 differences between all pairs in the sequence, taken modulo q 2 + q + 1 = 273, span the entire set [272], as guaranteed by Theorem 5.

9 Construction methods for Golomb rulers 243 Remark 3. This method is often referred to as the projective plane construction of Golomb rulers in the literature Symmetric Golomb Costas arrays construction [10, 9]. While studying the symmetry of Costas arrays, one of the present authors noticed that a particular subfamily of symmetric Golomb Costas arrays yielded a densely populated main diagonal [10, 9]. Theorem 6. Let q = p n be a power of a prime and g a primitive root in F(q 2 ). Then the set (23) S = {i [q 2 2] : g i + g qi = 1} contains exactly q integers which have distinct pairwise differences modulo q 2 1. Proof. The roots of (23) lie on the main diagonal of a Golomb Costas array (constructed using the primitive roots g and g q of F(q 2 ), the latter being a primitive root because (q, q 2 1) = 1), hence they form a Golomb ruler. Setting x = g i, (23) becomes (24) x q + x = 1, x F(q 2 )\{0, g}, and we just need to demonstrate that this equation has exactly q roots. To show this, we first observe that T(x) = x q + x is a linear map on the vector space F(q 2 ) over the field F(q): indeed, letting x, y F(q 2 ), a, b F(q), (25) T(ax + by) = (ax + by) q + (ax + by) = a q x q + b q y q + ax + by = = a(x q + x) + b(y q + y) = at(x) + bt(y), and the claim is proved. It follows that both the kernel and the image of T are subspaces of F(q 2 ). Assuming z Ker[T], we get z q = z. When p = 2, this implies that z F(q). When p > 2, (z 2 ) q = z 2 z 2 F(q); this implies that, choosing a w such that w 2 F(q), w / F(q), the roots z coincide with the set {w F(q 2 ) : w 2 F(q), w F(q)}. Hence, in both cases, dim(ker[t]) = 1 Ker[T] = q, which, in turn, implies that dim(im[t]) = 1 Im[T] = q. In particular, (24) either has q roots or no root; it is enough, then, to show that it has a root. Assume p > 2: restricting x F(q), (24) becomes 2x = 1 x = 1 2, hence (24) has a root. Assume p = 2: the previous trick fails to work, as, restricting x F(q) in (24), leads to the impossibility 0 = 1. Observing, though, that (T(x)) q = (x q + x) q = x 2q + x q = x q + x = T(x), which implies that Im[T] = F(q), we conclude that T(x) = 1 has a root, since 1 F(q). This completes the proof. These rulers are asymptotically optimal: they have q markings and their length is (at most) q 2 3.

10 244 Konstantinos Drakakis Example 5. Returning to the setting of Example 3, we need the corresponding table for the primitive root g 4 = x 4 = x + 1: x x x 3 + x 2 + x x 5 x 2 + x 6 x 3 + x 7 x 3 + x x 2 9 x 3 + x 2 10 x 2 + x x x 3 13 x 3 + x x 3 + x Comparing the two tables, we need to find the exponents i such that g i + g qi = 1 = x i + (1 + x) i. Exhaustive search yields the solutions i = 1, 2, 4, 8, hence the sequence 1, 2, 4, 8, or, equivalently, 0, 1, 3, 7, is a Golomb ruler of four markings and length 8. Note that this is the same ruler as the one produced in Example Dimitromanolakis construction [7]. The previous constructions work only when the number of markings is close to a (power of a) prime. The following construction [7] works always, but, unfortunately, is far from optimal; even in the original publication [7] it was only proposed as a toy example. Theorem 7. For any n N, and for a fixed c {1, 2}, the sequence (26) cnk 2 + k, k [n 1], forms a Golomb ruler. Proof. The proof for c = 2 is simple: start with the equation (27) 2n(x 2 + y 2 ) + (x + y) = a; dividing throughout by 2n, and given that 0 x + y 2n a (28) x 2 + y 2 =. 2n < 1, we get Substituting (28) back in (27), we get a (29) x + y = a 2n = a mod (2n), 2n and from (29), (28): (30) xy = 1 2 [(x + y)2 (x 2 + y 2 )] = 1 [ a ] (a mod (2n)) n (30), (29) show that {x, y} are the two roots of a polynomial of degree 2, hence determined in at most one way.

11 Construction methods for Golomb rulers 245 The proof for c = 1 is slightly more involved. We will now work with differences, and consider the equation (31) n(x 2 y 2 ) + (x y) = a, x > y; since 0 x y < n, it follows that (32) x 2 y 2 = a n, and by (31) and (32) we obtain a (33) x y = a n = a mod n. n Dividing (32) by (33), (34) x + y = x2 y 2 x y = a n a mod n, and finally (33) and (34) form a system of two equations and two unknowns that uniquely define x and y. This completes the proof. This ruler has n markings and its length is asymptotically cn 3, hence it is far from optimal. Example 6. Let us choose c = 1 and n = 12: the resulting sequence is , a Golomb ruler of 12 markings and length Optimal Golomb rulers In the absence of a theoretical result ascertaining the maximal number of markings for a Golomb ruler of a given length, or the minimal possible length for a Golomb ruler with a fixed number of markings, optimal Golomb rulers can only be discovered computationally, through exhaustive search. Actually, a better way to phrase this is that the only way to confirm the fact that a certain Golomb ruler is optimal is through exhaustive search: this is because, on many occasions, optimal Golomb rulers are actually constructed by the two construction methods based on finite fields (either Bose-Chowla or Singer). There are excellent resources on the web regarding optimal Golomb rulers, such as Wikipedia [23] or J. B. Shearer s page [20]. The former is more recently updated and contains optimal Golomb rulers for 24 markings or less (and possibly 25), while the latter also records the origins of the optimal Golomb rulers found: focusing on optimal Golomb rulers of more than 10 markings, we learn, in particular, that those for 11, 12, 14, 18, 19, 20, 21, 23, and 24 markings are obtained by the Singer construction (possibly by truncating the final marking), those for 17 and 22 markings by the Bose-Chowla construction, while only those for 13, 15, and 16 markings were new rulers found by computer search. If the candidate optimal Golomb ruler with 25 markings given in [23] is confirmed, it will also be Singerconstructed.

12 246 Konstantinos Drakakis (35) 5. A novel approach to the Bose-Chowla construction Turning our attention back to Theorem 4, we can reformulate (11) as (g i g) q = g i g g qi + ( g) q = g i g g qi g i = ( g) q g g qi g i = g q g, where the last inference is not obvious and requires distinguishing the two cases of even or odd characteristic, and verifying it holds true for both. Comparing this to (23), we see the expressions are similar, but not identical. Are they equivalent somehow? Can they be derived as special cases of a more general construction method? The answers are provided by the two theorems below: Theorem 8. Let q = p n be a power of a prime, g a primitive root in F(q 2 ), h F(q 2 )\F(q), and a F(q 2 ). Then the set (36) S a,h = {i [q 2 2] : ag i + h F(q)} has q elements, is a Golomb ruler modulo q 2 1, and, in fact, a cyclic shift of the Bose-Chowla construction (Theorem 4). Proof. To begin with, S a,h clearly has q distinct elements, since g is a primitive root and a 0. Furthermore, it represents a Golomb ruler: to see this, let i k S, k [4], such that i 1 > i 2, i 3 > i 4, i 1 + i 2 = i 3 + i 4, and let u k F(q), k [4] such that ag i k + h = u k. It follows that (37) g i1+i2 = g i3+i4 (u 1 h)(u 2 h) = (u 3 h)(u 4 h) u 1 u 2 h(u 1 + u 2 ) = u 3 u 4 h(u 3 + u 4 ) u 1 u 2 u 3 u 4 = h(u 1 + u 2 u 3 u 4 ). Assuming u 1 + u 2 u 3 u 4 0, we obtain that h F(q), a contradiction; hence, u 1 +u 2 = u 3 +u 4, which also implies through (37) that u 1 u 2 = u 3 u 4. It follows that {u 1, u 2 }, {u 3, u 4 } have both the same sum s and the same product r, hence they are obtainable as pairs of roots of the same polynomial of degree 2 over F(q), namely x 2 sx + r = 0, which either has two roots or no roots, hence {u 1, u 2 } = {u 3, u 4 }. This, in turn, implies that {i 1, i 2 } = {i 3, i 4 }. How many different sets S a,h are there? The key to count them is the observation that S a,h = S ta,th+h for h F(q) and t F(q). There are q 2 q possible values for h, namely the set F(q 2 )\F(q), while the (multi)set A h = {th+h : t F(q), h F(q)} also contains q(q 1) = q 2 q entries, possibly not distinct. We show now that A h is a set, namely that its elements are distinct: indeed, letting t 1, t 2 F(q) and h 1, h 2 F(q), so that t 1h + h 1, t 2h + h 2 A h, we obtain that (38) t 1 h + h 1 = t 2h + h 2 (t 1 t 2 )h = h 2 h 1 h = h 2 h 1 t 1 t 2 F(q) if t 1 t 2, which is a contradiction; hence t 1 = t 2, and consequently h 1 = h 2 as well. This proves our claim. Further, since A h F(q) h F(q), a contradiction, it follows that A h = F(q 2 )\F(q), which coincides with the possible range of h: in other words, for any two possible values h 1 and h 2 of h, we can find t and h such that h 2 = th 1 + h. Consequently, choosing h 2 = g, we see that the family of sets {S a,h : h F(q 2 )\F(q), a F(q 2 ) } coincides with the family of sets {S a, g : a F(q 2 ) }. But S a, g is just a cyclic shift of S 1, g modulo q 2 1, and the latter is the Bose-Chowla construction (see Theorem 4). This completes the proof.

13 Construction methods for Golomb rulers 247 Theorem 9. The construction proposed in Theorem 6 is just a cyclic shift of the Bose-Chowla construction (Theorem 4). Proof. We can rewrite (36) as (39) (ag i + h) q = ag i + h a q g qi + h q = ag i + h a q g qi ag i = h h q. Setting a q = a = h h q, we recover (23). Is this possible? Defining the operators H(x) = x q x and T(x) = x q + x, and repeating the argument in the proof of the Theorem 6, we find them to be linear transformations on the vector space F(q 2 ) over the field F(q). From the same argument we know that Ker[T] = q, while we see immediately that Ker[H] = F(q), whence also Im[T] = Im[H] = q: so all four sets are subspaces of F(q 2 ) of dimension 1. We just need to show that Ker[T] Im[H]. We can actually fare better and show Ker[T] = Im[H]: Assuming odd characteristic, (40) x Im[H] h : h q h = x x q = (h q h) q = h h q = x x Ker[T]. Since then Ker[T] and Im[H] have the same dimension, they must be equal. Regarding a, we can further infer that a F(q 2 )\F(q) and a 2 F(q): it is a square root of an element of F(q) not in F(q). Assuming even characteristic, it follows that T = H. The argument above can be repeated verbatim, disregarding all negative signs, and we obtain Ker[T] = Im[T] = F(q); this time, we obtain that a F(q). This completes the proof. 6. Golomb rulers with an arbitrary number of markings The construction methods for Golomb rulers presented in Section 3 apply whenever the number of markings is restricted to the set A = (Q 1) Q (Q + 1), where Q, Q are the primes and powers of primes, respectively; the only exception is Dimitromanolakis s method, which, though applicable for any number of markings, is, nonetheless, severely suboptimal to be of any practical value. How can we construct then a (reasonably dense) Golomb ruler with an arbitrary but preselected number of markings? 6.1. Truncated Golomb rulers. An simple way to construct a (reasonably dense) Golomb ruler of m / A markings, using the methods we have at our disposal, is to find the smallest a A such that a > m, construct the corresponding Golomb ruler using any among the Rusza-Lindström, Singer, or Bose-Chowla constructions, which are all asymptotically optimal, and then remove a m markings from the end of the ruler, i.e. truncate the ruler, to give it the required number of markings. Indeed, this construction is guaranteed to be 2-nearly optimal: according to Bertrand s conjecture (proved by P. Tchebychev), there is a prime between m and 2m, whence the length of the constructed Golomb ruler will be O(4m 2 ), hence the method is at most 2-nearly optimal: the lengths deviate from asymptotical optimality by a factor of at most 2, and actually much less in practice. We have thus shown that 2-nearly optimal constructions are possible for any number of markings; this estimate is not nearly as good as the one proved in [11] and repeated in Section 2 above, but it was obtained virtually without labor. In particular, this result implies Dimitromanolakis s method is (asymptotically) never preferable to truncating, and is, therefore, of purely academic interest only. A concern in the applicability of this method, however, is the fact that the elements

14 248 Konstantinos Drakakis of A are sparsely distributed in N, as a result of which the difference a m grows without bound A composition rule for Golomb rulers. The result below shows how we can combine two Golomb rulers with m 1 and m 2 markings, respectively, to obtain a new Golomb ruler with m 1 m 2 markings. Therefore, we can factorize m = a 1 a 2... a l, a i A, i [l], which is always possible as A contains powers of primes, preferably choosing the factorization with the fewest factors, construct the corresponding l Golomb rulers using the asymptotically optimal methods of Section 3, and compose them to obtain the desired ruler. Theorem 10. Let f i : [m i 1] [n i ], be a Golomb ruler modulo N i, i [l], such that (N i, N j ) = 1, 1 i < j l. Then the set { } l f i (x i ) l S = N : x i [m i 1], i [l], N := N i N i forms a Golomb ruler of m = l m i markings and of length n = N Proof. Let us consider the equation: (41) N l f i (x i ) N i N l f i (y i ) N i = a. Taking it modulo N i, i [l], we obtain the equations l n i N i. (42) f i (x i ) f i (y i ) a N i N mod N i, i [l]. The modular Golomb ruler property of each f i ensures that each of the equations above has at most one solution, hence the original equation (41) has at most one solution {x 1,..., x l, y 1,..., y l } for a given a. In particular, for a = 0 we obtain x i = y i, i [l], hence S has indeed m distinct elements, ranging from 0 to n. This completes the proof. Corollary 1. Let f i : [m i 1] [n i ], i [l] be Golomb rulers, and let N i 2n i +1, i [l] be such that (N i, N j ) = 1, 1 i < j l. Then the set { } l f i (x i ) l S = N : x i [m i 1], i [l], N := N i N i forms a Golomb ruler of m = l m i markings and of length n = N Proof. f i is modular modulo N i 2n i + 1, i [l], so Theorem 10 applies. l n i N i. Remark 4. The shortest Golomb ruler possible that can be produced by Corollary 1 for l = 2 is of length n = (2n 1 + 1)n 2 + (2n 2 + 1)n 1 4n 1 n 2. Assuming the original Golomb rulers satisfy n i m 2 i, i = 1, 2, their composition satisfies n 4n 1 n 2 = (2m 1 m 2 ) 2 = (2m) 2. The composition does not preserve, then, asymptotical optimality, but is still 2-nearly optimal.

15 Construction methods for Golomb rulers 249 Example 7. Assume we wish to construct a reasonably dense Golomb ruler with 187 markings. We factorize 187 = 11 17, and we can apply e.g. the Bose-Chowla construction to obtain two modular Golomb rulers of lengths at most = 118 and = 286, respectively. Then we can compose them to obtain a Golomb ruler of 187 markings and of length at most = Example 8. Suppose we want to create a Golomb ruler of 33 markings, by composing rulers of three and 11 markings, respectively. We will use the rulers 0, 1, 4, 13, 28, 33, 47, 54, 64, 70, 72 and 0, 1, 3, which are modular modulo 145 and 7, respectively. The composed Golomb ruler is S = {0, 7, 28, 91, 145, 152, 173,196, 231, 236, 329, 341, 376, 378, 435, 442, 448, 463, 474, 490, 504, 523, 526, 593, 631, 635, 649, 666, 764, 813, 883, 925, 939} of 33 markings and of length Summary and conclusion In this work we have collected together the main construction methods for Golomb rulers published in the literature, along with their proofs and several examples. We saw that three methods (Bose-Chowla, Rusza-Lindsröm, Singer) are asymptotically optimal, one (Erdös-Turan) is 2-nearly optimal, and one (Dimitromanolakis) is severely suboptimal and intended as a toy example only. We also discussed briefly optimal Golomb rulers, and we observed that the two methods that make use of the structure of finite fields (Bose-Chowla, Singer) very frequently produce optimal Golomb rulers. A third construction method based on finite fields was proposed in the mathematical literature, whereby asymptotically dense Golomb rulers were obtained by the main diagonals of a special subfamily of symmetric Golomb rulers. We showed here that this method actually produces cyclic shifts of the Bose-Chowla construction, hence Bose-Chowla constructed Golomb rulers are obtainable as the main diagonals of Golomb Costas arrays. This observation links asymptotically optimal Golomb rulers to Costas arrays. The disadvantage of these construction methods is that they work only for a number of markings that is either a power of a prime, 1 plus the power of a prime, or a prime minus 1. To obtain Golomb rulers for a different number of markings, one possibility is to construct a ruler with more markings using one of the methods and truncate it. Alternatively, we saw that two Golomb rulers can be composed to yield a longer Golomb ruler whose number of markings is the product of the numbers of markings of the two constituent rulers. Acknowledgements This material is based upon works supported by the Science Foundation Ireland under Grant No. 05/YI2/I677, 06/MI/006 (Claude Shannon Institute), and 08/RFP/MTH1164. References [1] W. C. Babcock, Intermodulation interference in radio systems/frequency of occurrence and control by channel selection, Bell System Technical Journal, 31 (1953),

16 250 Konstantinos Drakakis [2] F. Biraud, E. Blum and J. Ribes, On optimum synthetic linear arrays with application to radioastronomy, IEEE Transactions on Antennas and Propagation, 22 (1974), [3] R. C. Bose, An affine analogue of Singers theorem, Journal of the Indian Mathematical Society, 6 (1942), [4] R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Commentarii Mathematici Helvetici, 37 ( ), [5] J. P. Costas, Medium constraints on sonar design and performance, Technical Report Class 1 Rep. R65EMH33, GE Co., (1965). [6] J. P. Costas, A study of detection waveforms having nearly ideal range-doppler ambiguity properties, Proceedings of the IEEE, 72 (1984), [7] A. Dimitromanolakis, Analysis of the Golomb Ruler and the Sidon set problems, and determination of large, near-optimal Golomb rulers, Diploma thesis, Department of Electronic and Computer Engineering, Technical University of Crete, Available at [8] K. Drakakis, A review of Costas arrays, Journal of Applied Mathematics, [9] K. Drakakis, R. Gow and L. O Carroll, On some properties of Costas arrays generated via finite fields, IEEE CISS, [10] K. Drakakis, R. Gow and L. O Carroll, On the symmetry of Welch- and Golomb-constructed Costas arrays, Discrete Mathematics, 309 (2009), [11] P. Erdös and P. Turán, On a problem of Sidon in additive number theory and some related problems, Journal of the London Mathematical Society, 16 (1941), Followed by Addendum (by P. Erdös), ibid., 19 (1944), [12] R. García, L. Díez, J. A. Cortés and F. J. Cañete, Mitigation of cyclic short-time noise in indoor power-line channels, IEEE International Symposium on Power Line Communications and Its Applications, 2007, [13] S. Golomb, How to number a graph, in Graph Theory and Computing (ed. R. C. Read), Academic Press, New York, (1977), [14] S. Golomb, Algebraic constructions for Costas arrays, Journal Of Combinatorial Theory Series A, 37 (1984), [15] S. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proceedings of the IEEE, 72 (1984), [16] B. Lindström, An inequality for B 2 -sequences, Journal of Combinatorial Theory, 6 (1969), [17] B. Lindström, Finding finite B 2 -sequences faster, Mathematics of Computation, 67 (1998), [18] J. P. Robinson and A. J. Bernstein, A class of binary recurrent codes with limited error propagation, IEEE Transactions on Information Theory, 13 (1967), [19] I. Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arithmetica, LXV (1993), [20] J. B. Shearer s webpage on Golomb rulers: [21] S. Sidon, Ein Satz über trigonometrische polynome und seine anwendungen in der theorie der Fourier-Reihen, Mathematische Annalen, 106 (1932), [22] J. Singer, A theorem in finite projective geometry and some applications to number theory, Transactions of the American Mathematical Society, 43 (1938), [23] Wikipedia s entry on Golomb rulers: Received January 2009; revised July address: Konstantinos.Drakakis@ucd.ie