Bh Sets as a Generalization of Golomb Rulers

A set of positive integers <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> is called a Golomb ruler if the difference between two distinct elements of <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> are different, equivalently if the sums of two elements are different (<inline-formula> <tex-math notation="LaTeX">$B_{2}$ </tex-math></inline-formula> set, Sidon set). An extension of this concept is to consider that the sum of <inline-formula> <tex-math notation="LaTeX">$h $ </tex-math></inline-formula> elements in <inline-formula> <tex-math notation="LaTeX">$A $ </tex-math></inline-formula> are all different, except for permutation of the summands, with <inline-formula> <tex-math notation="LaTeX">$h \geq 2 $ </tex-math></inline-formula>, in this case it is said that <inline-formula> <tex-math notation="LaTeX">$A $ </tex-math></inline-formula> is a set <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula>, the length of <inline-formula> <tex-math notation="LaTeX">$A $ </tex-math></inline-formula> is given by <inline-formula> <tex-math notation="LaTeX">$\ell (A)\,\,= \max A- \min A $ </tex-math></inline-formula>. One problem associated with this type of set is that of the optimal dense <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> sets, that is, determining the greatest cardinal of a set <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> contained in the integer interval <inline-formula> <tex-math notation="LaTeX">$\left [{1, N }\right] $ </tex-math></inline-formula>, for this defines the function <inline-formula> <tex-math notation="LaTeX">$F_{h} (N)~$ </tex-math></inline-formula>. Another problem that can be associated is the optimally short <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> sets, that is, finding a shorter <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> set with <inline-formula> <tex-math notation="LaTeX">$m $ </tex-math></inline-formula> elements, for which the <inline-formula> <tex-math notation="LaTeX">$G_{h} (m)~$ </tex-math></inline-formula> function is defined. In this paper we are going to prove that these two problems are inverse, that is, that the functions <inline-formula> <tex-math notation="LaTeX">$G_{h} (m)~$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$F_{h} (N)~$ </tex-math></inline-formula> have inverse relationships. Furthermore, the asymptotic behavior of the <inline-formula> <tex-math notation="LaTeX">$G_{h} (m)~$ </tex-math></inline-formula> function is studied, obtaining some upper and lower bounds, we also obtain tables of <inline-formula> <tex-math notation="LaTeX">$B_{3}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$B_{4}$ </tex-math></inline-formula> near-optimal up to <inline-formula> <tex-math notation="LaTeX">$m = 31$ </tex-math></inline-formula>.


I. INTRODUCTION
Intermodulation interference is the combining of several signals in a nonlinear device, producing new, unwanted frequencies. The intermodulation between frequency components will form additional components such as harmonic frequencies (integer multiples), sums, and differences of the original frequencies. These new frequencies are called the intermodulation product.
W. C. Babcock in his work on assigning radio frequencies ( [1], [3]) to avoid third and fifth-order of intermodulation interference, formulated the following problems.
1) For any given m, find integers a 1 < a 2 < · · · < a m so as to the equation a r + a s − a t = a u does not have different solutions from the trivial one.
The associate editor coordinating the review of this manuscript and approving it for publication was Sun-Yuan Hsieh .
2) For any given m, find integers a 1 < a 2 < · · · < a m so as to the equation a r + a s + a t − a u − a v = a w doesn't have different solutions from the trivial one. Hence the need to study sets with the property that all the sums of two elements are different and sets with the property that all sums of three elements are different (B 3 set).
A set of non-negative integers in which all the differences of two elements different or equivalently sums of two elements are distinct is called Golomb ruler; the elements of this ruler are called marks.
The Golomb rulers are important by their applications in different fields of engineering and communications, see [1], [3], [9]. Another application of Golomb Rulers is in the field of coding theory for find to optimal optical orthogonal Codes [19]. Sets of rulers are used to generate self-orthogonal codes that play an important role in communications [17], the optimal Golomb Rulers are used for the FWM Crosstalk Elimination in WDM Systems [20].
Definition 1: A Golomb ruler is a set of integers A = {a 1 , a 2 , . . . , a m } with a 1 < a 2 < · · · < a m , in which for each positive integer d there are not more than one solution of the equation d = a i − a j , where i > j, its number of elements is called order and the largest distance between two elements of the ruler is called length, denoted (A), So.
Since the concept of the Golomb ruler is invariant under translations, it is possible to assume that the minimum value is a 1 = 0 and the length is a m .
The fundamental problem in the study of the Golomb rulers is to find the shortest rulers for a certain number of marks; equivalently investigate the following function: We say that a Golomb ruler of order m is optimal if it has the shortest length possible (optimally short). For example, the ruler given in (1) where m = 15 is a optimal Golomb ruler with length 151 and G(15) = 151. Currently, there are also optimal Golomb rules where 2 ≤ m ≤ 27 marks [8], [16] and there is an ongoing search for an optimal 28-marks rule. Dimitromanolakis [7] proved computationally in 2002 that G(m) ≤ m 2 , for every m ≤ 65000, and he conjectured that this is true for all integer m.
In [18], Rokicki and Dogon obtain near-optimally Golomb rules for values for m up to 40000, they use some new ideas and improve the existing algorithms.
We can find a trivial lower bound counting the number of distinct differences of a Golomb ruler with m elements, so Some researchers achieved best the trivial lower bound, their results are the followings ). Atkinson et al. [1] it is conjectured that for all m is possible that On the other hand in [1] Atkinson et al. studied a generalization of Golomb ruler (when the sum of 3 elements in A are all different), these are called B 3 sets (for another generalization see [13]- [15]), furthermore, this concept can be generalized for sums of h elements i.e. an integers set A = {a 1 , a 2 , . . . , a m }, is called a B h set, if the sum of h elements in A are all different, except for permutation of the summands, with h ≥ 2. One problem associated with this type of sets is optimally dense B h sets, that is, determining the greatest cardinal of a set B h contained in the integer interval [1, N ], for this defines the function F h (N ) . Another problem that can be associated is the optimally short B h sets, that is, finding a shorter B h set with m elements, for which the G h (m) function is defined, in [1] Atkinson et al. presented near-optimally short B 3 sets with m ≤ 18 marks and in [12] Lam and Duan presented optimal B 3 sets with m ≤ 7 and B 4 with m ≤ 6. In this paper we are going to prove a generalization of the results given [7], i.e. that the functions G h (m) and F h (N ) have inverse relationships. Furthermore, the asymptotic behavior of the G h (m) function is studied, obtaining some upper and lower bounds.
In Section II, we introduce the concept of B h set, we considered two fundamental problems about these sets and we present some properties, constructions, and known results. In section III we show that the maximal functions F h (N ) and G h (m) have inverse relationships, generalizing the results of [7]. In Section IV, we present upper and lower bounds for the function G h (m) and we generalize the results presented in [22] for B h sets and arbitrary module.

II. GENERALIZED GOLOMB RULERS
Now we present the formal definition of a B h set.
Definition 3: Let (G, +) be an abelian group, A be a subset of G and h ≥ 2 an integer. A is a B h set in G if the sums of h elements of A are all different i.e. for all x ∈ G, the equation with a i j ∈ A, it has at most a single solution in A, except for permutations of the summands.
If G = Z N the set of integers modulo N , then A is called a modular B h set. Notice that a modular B h set is also a B h set in Z, analogously we can define length, marks, order and the G h (m) function.

A. TWO FUNDAMENTAL PROBLEMS
The main problem is to an find optimally dense and optimally short B h set, i.e. finding answers to the following two optimization questions. Let A ⊂ Z be a B h set and m, n ∈ Z + .
If h = 2, then G 2 (m) = G(m) and exact values for 1 < m < 27 are known for this function. On the other hand, Caicedo, Martos and Trujillo proved in [4] that: For h = 3, Atkinson, Santoro and Urrutia proved in [1] that: Furthermore, in [12] Lam and Duan find optimal B 3 sets with m marks for m ≤ 7 and B 4 for m ≤ 6.

Problem 2 (Optimally Dense B h sets):
This problem is related to the estimation of the following function: For h = 2 Cilleruelo proved in [4], [5] that: There is a trivial upper bound, it is given by: Bose and Chowla [2] conjecture that lim In the case of h = 3 and h = 4, in [11] it is proved that: In [11] it is mentioned that it has Been Green proved that: (1)).
From where it can be deduced

B. CONSTRUCTIONS
Three optimal constructions for B h sets are known, these are the Bose-Chowla type, generalized Singer type, and Gómez-Trujillo type, we are going to quickly describe each one of them since they provide us with bounds for the extreme functions.
Let F be a field, E be an algebraic extension of degree h on F, and α an algebraic element of degree h on F, if E * is the multiplicative group of E, then, the set is a B h set in E * see [10]. Now if F is a finite field then F = F q , with q prime power, α an algebraic element of degree h on F q and θ a primitive element of the extension field F q h on F q , the set from where Bose and Chowla [2] proved that.
is a B h set modulo (q h − 1) with q elements. Example 8: Let F 5 be finite field, and θ = α be a primitive root of the polynomial x 3 + 3x + 2 on F 5 . Then F 5 3 is an extension field with degree 3 on F 5 and the elements of F *

3
can be written as powers of θ.
The set  [2] can be seen as a consequence of the Bose-Chowla-type construction.
Let q be a prime power, F q h+1 be a finite field with primitive element θ and α an algebraic element of degree h + 1 on F q . If α = θ, the set θ + F q is a set B h+1 with q elements in the multiplicative group of F q h+1 . By Theorem 7, we can obtain a in [2] Bose and Chowla proved that S ∪ {0} is a B h set in Z N q , + with q + 1 elements.
Theorem 9 (Singer generalized): Let q be a power prime and h ≥ 2 an integer, there are q + 1 integers  a 1 , a 2 , . . . , a q , a q+1 such that all sums a j 1 + a j 2 + · · · + a j h , is a B h set in Z p × Z p h −1 by Theorem 11. Furthermore, by the Chinese remainder Theorem we obtain a B h set modulo p h −p with p elements, given by Example 12: Let p = 7, θ be a primitive element on F 7 with minimal polynomial p(x) = 2x 2 + x + 3, then by Theorem 9, the set

III. RELATIONSHIPS BETWEEN MAXIMAL FUNCTIONS
For the sets B 2 there is an inverse relationship between the functions G(m) and F 2 (N ) , which were proved in 2002 by Dimitromanolakis [7]. In this section we prove a generalization of the results of [7] for the case of the sets B h . To find equality relations between the functions, let us assume that the exact value of F h (N ) or G h (m) is known, from these values we will find some relations between them.
Lemma 13: If n, m ∈ Z, then, From the previous corollary we can deduce some exact values for the function F 3 (N ) and F 4 (N ) from the known ones for G 3 (m) and G 4 (m) given in [12] see Table 1. Proof: Analogous to Lemma 15. Using this lemmas, we prove that. Theorem 17: Suppose l(n) and u(n) are well-defined and there are inverse functions l −1 (n) y u −1 (n) inside an integer interval I ⊂ N. If Proof: Suppose that F h (n) > l(n), by Lemma 15 G h (l(n)) ≤ n − 1 therefore: On the other hand, if F h (n) < u(n) then G h (u(n)) ≥ n − 1 by Corollary 16, so we have Let us now consider the case in which we know some bound for G h (m) and see what results can be obtained for F h (n).
On the other hand, if G h (m) < u(m), then F 2 (u(m)) ≥ m by Lemma 18, hence: The theorems 17 and 20 are important because, in addition to proving the inverse relationship between the two problems, it allows finding upper and lower bounds for the function G h (m) from the known bounds for F h (N ) , this is done in the next section.

IV. UPPER AND LOWER BOUNDS
In this section, we are going to properly construct a short B h set with p elements and thus find an upper bound for G h (p) with p prime. On the other hand, to find a lower bound we are going to make use of the inverse relationship that exists between the functions G h (m) and F h (n) presented in Theorem 20.
With the constructions of the Theorems 9, 7 and 11 we can obtain upper and lower bounds for the functions G h (m) if m is a prime number and F h (N ) see Table 2.

A. AN UPPER BOUND
Using a technique initially presented by Zhang [22], where it is allowed to construct Golomb rulers with a suitable length from modular rulers, we can improve the upper bound for the function G h (m) presented in Theorem 27. In his paper, Zhang only considers Golomb rulers obtained from Bose construction. In this paper, we generalize the results presented in [22] for B h sets and arbitrary modules.
To find an upper bound of the function G h (m) we must build a B h set with m marks and (A) = w. In this case, G h (m) ≤ w.
Definition 21: Let A = {a 1 , a 2 , . . . , a m , a m+1 } be a B h set with m + 1 marks and 1 = a 1 < a 2 < · · · < a m+1 . We define D(m) by Note that there are, at least m distinct consecutive differences because A is a B h set, so  Proof: Reasoning by contradiction, without loss of generality, suppose that there are two sums of h elements that coincide as follows: As a consequence of (6) and the constructions of B h sets, we have.
Theorem 27: Let p be a prime number, q be a prime power, we have:

VOLUME 9, 2021
Proof: By Gómez-Trujillo construction, we have a B h set with q marks, modulo q h − q, from Theorem 25 we have a set B, where B is a B h set and (B) = q h −q−D(q), therefore Analogously we can now prove 2 and 3, with Singer generalized an Bose-Chowla constructions.
We can now rephrase this theorem for h = 3.
Corollary 28: Let p be a prime number, q be a prime power, then we have: And for h = 4 we can have. Corollary 29: Let p be a prime number, q be a prime power, then we have: In Lemma 4 we proved that every B h set in Z with m elements is a B h sets in Z N , where N = ha m + 1. So we can prove that v γ ,h (m) ≤ hG h (m) + 1.

B. A LOWER BOUND
To find a lower bound of the function G h (m) , we are going to use the inverse relationship between the functions F h (n) and G h (m) given in Theorem 20. In the following theorem, we obtain a lower bound for G h (m) using a known upper bound for F h (N ) .
Theorem 31: Let m be a positive integer, then then u −1 (n) = n h hḣ! , therefore by Theorem 17 we have the result Now for the case h = 3 we are going to use an upper bound given in [11] for the function F 3 (n) .
Theorem 32: Let m be a positive integer, then Proof: We know by 3 that F 3 (n) ≤ 3 24 5 n + 2, Let then there is u −1 (n) and its given by then by the Theorem 17 we have from which the desired result is obtained. The lower bound obtained in Theorem 32 represents an improvement for large values of m, with respect to the bound given by Atkinson et al. [1], which is given by G 3 (m + 1) ≥ 10 57 m 3 . Indeed, it can be observed that starting from m = 37, the bound of the previous theorem is better than that given by Atkinson, Santoro and Urrutia.
For h = 4 we have. from where the desired result is obtained.

V. CONCLUSION
In this paper we find inverse relationships between the functions F h (N ) and G h (m) generalizing the results of [7]. Additionally, in this paper, we make a study of the asymptotic behavior of the function G h (m) obtaining an upper and lower bound. On the other hand, some questions that can be addressed in future work, which we consider interesting to approach the following problems. • An inverse relationship is demonstrated between the functions F h (n) and G h (m) , results that generalize those given by Dimitromanolakis in [7]. Can these results be used to improve the existing bounds? Can the bounds of the functions be improved? F 3 (N ), F 4 (N ), G 3 (m) y G 4 (m)?
• It is possible to improve the bounds for the function F h (N ) , is there the limit lim N →∞ F h (N ) N 1/h ?
• To study the asymptotic behavior of the function v δ,g (m) with g > 1 and obtain new optimal bounds.
• To consider analogous problems for the modular case and for sums sets.

APPENDIX ANOTHER NUMERICAL RESULTS
In [1] Atkinson et al. obtained the suboptimal B 3 sets to m = 18 and in [12] Lam and Duan presented optimal B 3 sets to m = 7 y B 4 sets to m = 6. Using the constructions of B h sets Singer Generalized(SG), Bose-Chowla(BC) and Gómez-Trujillo(GT), truncating and the Lemma 4, we have found, as shown in Table 3 and Table 4, the suboptimal B 3 and B 4 sets for m < 31. In these tables we present the construction used and the prime number or prime power used.