Almost Difference Sets From Singer Type Golomb Rulers

Let <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> be an additive group of order <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula>. A <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-element subset <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called a <inline-formula> <tex-math notation="LaTeX">$(v, k, \lambda, t)$ </tex-math></inline-formula>-almost difference set if the expressions <inline-formula> <tex-math notation="LaTeX">$g-h$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula>, represent <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> of the non-identity elements in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> exactly <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> times and every other non-identity element <inline-formula> <tex-math notation="LaTeX">$\lambda + 1$ </tex-math></inline-formula> times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. 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. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.


I. INTRODUCTION
Difference sets are a well-known class of mathematical objects used in the construction of designs and other combinatorial structures. A k-subset D in an additive group G of order v is called a (v, k, λ) difference set DS (in G) if δ D (x) = λ for every nonzero element of G, where δ D (x) is the difference function defined by The difference function counts the number of representations of x in the form Many groups do not have DSs for any parameters k and λ, but do have structures that are very close to DSs, which motivates the following definition.
A k-element subset D in an additive group G of order v is said to be a (v, k, λ, t)-almost difference set ADS (in G) if δ D (x) takes on the value λ altogether t times and λ + 1 altogether v − t − 1 times as x ranges over G \ {0}. This is, δ D (x) = |(D + x) ∩ D| = λ or λ + 1, The associate editor coordinating the review of this manuscript and approving it for publication was Sun-Yuan Hsieh .
Note that almost difference sets are a generalization of difference sets (when t = 0 or t = v − 1). Moreover, for an almost difference set D with parameters (v, k, λ, t), its complement G \ D is also an almost difference set with parameters (v, v − k, v − 2k + λ, t). An almost difference set D is called abelian or cyclic if the group G is abelian or cyclic, respectively. Almost all difference sets are interesting combinatorial objects that have several applications in many engineering areas. In coding theory, they can be employed, to construct cyclic codes [13]. Additionally, in cryptography, they can be used to construct functions with optimal nonlinearity [8], [11]. Finally, for CDMA communications, some cyclic almost difference sets yield sequences with optimal autocorrelation [1], [15], [16].
Different definitions of an almost difference set were independently developed by Davis and Ding in the early 1990s [9], [11], [12]. Let G be an additive group of order mn and N a subgroup of G of order n. A k-subset D in G is called a (mn, n, k, λ 1 , λ 2 ) divisible difference set DDS (in G) provided that the difference function δ D (x) defined above takes on the value λ 1 for each nonzero x ∈ N and takes on the value λ 2 for each nonzero x ∈ G \ N . That is, for each If λ 1 = 0, D is called a relative difference set RDS, and N is called the forbidden subgroup. Davis [9] called a divisible difference set an almost difference set in the case where |λ 1 − λ 2 | = 1. Davis defined this special class of almost difference sets due to its relationship with symmetric difference sets. Another kind of almost difference sets were defined by Ding [11], [12] for the study of cryptographic functions with optimal nonlinearity. He considered the current definition when t = v−1 2 . Number theoretic constraints can be applied to show that some groups cannot contain ADSs with certain parameters [32]. One example is that (v − 1)λ + t = k(k − 1) must hold for any ADS. Other criteria can be discovered by examining the quotient groups of the original group. Despite the effectiveness of these techniques, no general existence criterion is known to determine exactly which groups contain ADSs [7]. Many known construction methods exist for almost difference sets [1], [3], [14], [16]- [18], [32]. These constructions come from: difference sets, cyclotomic classes of finite fields, support of some functions, binary sequences with three-level autocorrelation, or larger product group. For a good survey of almost difference sets, the reader is referred to [24].
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 exists at most 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 According to [5], the Golomb rulers were first discovered by W.C. Babcock in 1950 when he investigated the intermodulation distortion. However, Golomb rulers derive their name from Professor Solomon W. Golomb, one of their greatest pioneers.
The concept of the Golomb ruler is invariant under linear applications; if A = {a 1 , a 2 , . . . , a m } is a Golomb ruler, then the set x · A + y := {xa 1 + y, x a 2 + y, . . . , x a m + y}, is a Golomb ruler, for all x, y ∈ Z, with x = 0. Thus, 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: A Golomb ruler of order m is optimal if it has the shortest possible length. For example, {0, 4, 20, 30, 57, 59, 62, 76, 100, 111, 123, 136, 144, 145, 151} is an optimal Golomb ruler with length 151 and G(15) = 151. Currently, there are also optimal Golomb rulers where 2 ≤ m ≤ 27 marks [8], [14] and there is an ongoing search for an optimal 28-marks rule. For more on Golomb rulers, their generalizations, and applications, see [25], [26], and the references in them.
In this paper, we use Singer type Golomb rulers (which are difference sets with λ = 1, or almost difference set with λ = 0 and t = 0) to construct new families of almost difference sets. These constructions are new, as far as we are aware of. The first construction yields (N /3, q, 2, 2(q − 1))-ADSs in cyclic groups of order N /3, where N = q 2 + q + 1 and q ≡ 1 mod 3 is a prime power greater than 4. This construction uses homomorphic projection. The second construction is obtained by adding a new element to the Golomb ruler and yields (q 2 + q + 1, q + 2, 1, (q − 2) (q+1))-ADSs in cyclic groups of order q 2 +q+1 for all prime power q. The third construction is obtained by removing an element of the Golomb ruler and yields (q 2 + q + 1, q, 0, 2q)-ADSs in cyclic groups of order q 2 + q + 1 for all prime power q. The latest constructions follow the idea proposed in [1].
Another contribution of this paper is related to t-adesign, which was defined in [14]. Let D = (P, B, I) be an incidence structure with v ≥ 1 points and b ≥ 1 blocks, where every block has size k. If every subset of t points of P is incident with either λ or λ+1 blocks of B, then D is called a t-(v, k, λ) adesign, or simply t-adesign The following lemma was established in [23] and provides a relationship between almost difference set and t-adesign.
Using the above lemma and the almost difference sets constructed in this paper, we give constructions of 2-adesigns.

II. ALMOST DIFFERENCE SETS VIA SINGER TYPE GOLOMB RULERS
In this section, we describe three new constructions of almost difference sets from Singer type Golomb rulers. These constructions can generate infinitely many almost difference sets in Z n for appropriate values of n. First, we present the formal definition of a Golomb ruler.
Definition 1: Let G be an additive abelian group, and A be a subset of G. A is a Golomb ruler From the definition, it is easy to verify that a modular Golomb ruler is a (v, k, 0, t) almost difference set.
If A is a subset of the additive group G and n ∈ N, then A mod n := {a mod n : a ∈ A}; The following construction of modular Golomb ruler is due to Singer [28]. He was not working on number theory but on finite projective geometry when he found this construction. Currently this construction can be described as follows [4], [20].
Proposition 1: Let q be a prime power, θ be a primitive element of the finite field F q 3 , α ∈ F q 3 be an element with cubic minimal polynomial over F q , and S = {log θ (α + u) : u ∈ F q }. Then S 0 = S mod (q 2 + q + 1) ∪ {0}, is a Golomb ruler in Z q 2 +q+1 , with q + 1 elements. Moreover, Remark 1: S 0 is a (q 2 + q + 1, q + 1, 1) difference set. Example 1: Let q = 7. If θ is a root of the primitive polynomial x 3 + 4x 2 + 4x + 4 over F 7 and α = θ. Then Taking the discrete logarithm of A in base θ yields the set The following theorem shows how to construct an almost difference set from a Singer type Golomb ruler using homomorphic projection. , q, 2, 2(q − 1))-ADS. Proof: According to Singer's construction, for every prime power q, there is a Golomb ruler S 0 in Z q 2 +q+1 , with q + 1 elements, particularly for q ≡ 1 mod 3.
On the other hand, by (Remark 1) there are unique elements s i , s j , s k , s l , s t and s r in S 0 such that x = s i − s j , x 1 = s k − s l , and x 2 = s t − s r , so, by (1) this is, As ϕ(s 1 ) = ϕ(s 2 ) = d 1 then for 3 ≤ j ≤ q + 1, the 4(q − 1) pairwise distinct elements therefore, there are 2(q − 1) distinct elements of Z q 2 +q+1 , q, 2, 2(q − 1))-ADS. The following proposition shows how to construct an almost difference from a difference set by adding an element. 16 ) difference set in G, and let d ∈ G \ D. If 2d cannot be written as the sum of two distinct elements of D, then D ∪ {d} is a (v, v+3 4 , v−5 16 , v−1 2 ) almost difference set in G, see [1].
Using the same idea of Proposition 2, we obtain the following result.
Proof: According to Singer's construction, for every prime power q, there is a Golomb ruler S 0 in Z q 2 +q+1 . In particular, S 0 is a (q 2 + q + 1, q + 1, 1)-DS. Then, the result follows applying Theorem 2 with a suitable element in Z q 2 +q+1 \ S 0 . Remark 3: Two elements cannot be added to S 0 in Theorem 2 to obtain a (q 2 + q + 1, q + 3, 1, t) almost difference set. Indeed, let x 1 and x 2 be two distinct elements in Z q 2 +q+1 \ S 0 and D = S 0 ∪ {x 1 , x 2 }. As x 1 = x 2 , then x 1 − x 2 ∈ S 0 S 0 (see Remark 1), so for some s 1 , s 2 ∈ S 0 (s 1 = s 2 ). As y = 0, then y ∈ S 0 S 0 . Therefore, y can be written in three different ways as differences of elements in D.
Example 6: The set {0, 1, 11, 19, 26, 28} is a Golomb ruler in Z 31 . By adding 9, and 24, we obtain the set D = {0, 1, 9, 11, 19, 24, 26, 28}. Note that 9 and 24 cannot be written as the sum of two distinct elements of D, but the element 29 in Z 31 can be written as 24 − 26 ≡ 9 − 11 ≡ 26 − 18. Other elements in Z 31 can also be written in three different ways as differences of elements in D; for example 8.

C. CONSTRUCTION 3
The following proposition shows how to construct an almost difference set from a difference set by removing an element.
Proposition 3: Let D be a (v, v+3 4 , n+3 16 ) difference set in G, and let d ∈ D. If 2d cannot be written as the sum of two distinct elements of D, then D \ {d} is a (v, v−1 4 , v−13 16 , v−1 2 ) almost difference set in G, see [1].
Using the same idea of Proposition 3, we obtain the following result.
, then 2d cannot be written as a sum of two distinct elements of D; therefore Corollary 2: There is a (q 2 + q + 1, q, 0, 2q)-ADS in Z q 2 +q+1 , for all prime power q.
On the other hand, there are some questions that can be addressed in future work; we consider it interesting to approach the following problems: 1) To study the structure, properties, and applications of the almost difference sets constructed in this paper. 2) Let Z v be the residue class ring module v and t be a divisor of v. Moreover, let S be a difference set in Z v , ϕ : Z v → Z v t be the homomorphism defined by ϕ(a) ≡ a mod v t , and D = ϕ(S). For which values of t do the set D form an almost difference set?
3) Is there some infinite family of almost difference sets with parameters (n, k, 2, t), and different from Theorem 1? Is there some infinite family of almost difference sets with parameters (n, k, 1, t)?