A recursive construction for skew Hadamard difference sets

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group G contains a skew Hadamard difference set, then G must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it. Mathematics Subject Classifications: 05B10, 05E30, 11T22


Introduction
Let G be an additively written group, let G * = G \ {0 G }, and let D be a subset of G. We say that D is a difference set if the list of differences "x − y, x, y ∈ D, x = y" represents every element of G * exactly λ times. In this paper, we are concerned with difference sets in abelian groups. We say that a difference set is skew Hadamard if D is a skew-symmetric (|G| − 1)/2-subset of G, i.e., D ∪ −D = G * and D ∩ −D = ∅, where −D = {−x : x ∈ D}. Two difference sets D 1 and D 2 in an abelian group G are said to be equivalent if there is a group automorphism σ of G and an element b ∈ G such that σ(D 1 ) = D 2 + b.
Let D be a skew Hadamard difference set in G, and let D + x = {y + x : y ∈ D} for x ∈ G. It is known that the collection Dev(D) = {D + x : x ∈ G} forms a symmetric 2-design, called a Hadamard design, which gives rise to a skew Hadamard matrix of order |G|+1. Thus, the problems on the existence and classification of skew Hadamard difference sets are well-rooted in design theory.
We will use the following well-known property of skew Hadamard difference sets later. The primary example of skew Hadamard difference sets is the classical Paley difference set in the additive group of the finite field F q (q ≡ 3 (mod 4)), which consists of all nonzero squares of F q . The automorphism group of the design developed from the Paley difference set was determined by Carlitz [1] and Kantor [10]. The Paley difference set was the only known example in abelian groups for many years. Therefore, many researchers had believed that up to equivalence the Paley difference sets are the only skew Hadamard difference sets in elementary abelian groups. In 2006, Ding and Yuan [5] disproved this conjecture by giving counterexamples of skew Hadamard difference sets in (F 3 5 , +) inequivalent to the Paley difference set. This discovery re-energized the research on skew Hadamard difference sets. On the other hand, the following conjecture is also known.

Conjecture 2.
If an abelian group contains a skew Hadamard difference set, then the group is necessarily elementary abelian.
This conjecture is still open in general while an exponent bound on groups containing skew Hadamard difference sets was studied in [4]. In this paper, we make some progress on this conjecture.
Many constructions for skew Hadamard difference sets in (F q , +) have been known in the past two decades as listed in Table 1. They are classified into three types: (1) Constructions as images of polynomials over F 3 ; (2) product constructions in F 3 q ; (3) constructions based on cyclotomy. In particular, the construction by Muzychuk [14] and its generalization by Chen-Feng [2] are very powerful; indeed their constructions yield many inequivalent skew Hadamard difference sets but the group is limited to (F n q , +) with n = 3. For large n > 3, Feng-Xiang's skew Hadamard difference sets [8,13] are the only known class containing infinitely many examples inequivalent to the Paley difference sets. Thus, the problem on whether there exists a skew Hadamard difference set inequivalent to the Paley difference set in (F q , +) for every odd prime power q ≡ 3 (mod 4) is still unsolved.
The purpose of this paper is to give a recursive construction for skew Hadamard difference sets in abelian groups. As far as the author knows, no recursive construction was known while "recursive-like" product constructions were known. The construction given by Muzychuk [14] and its generalization by Chen-Feng [2] needs one skew Hadamard difference set in F q and one "vertically balanced" Paley type partial difference set in F 2 q to construct a skew Hadamard difference set in (F 3 q , +). On the other hand, Chen-Feng's construction [3] needs an "Arasu-Dillon-Player" difference set in F * q n /F * q to construct a skew Hadamard difference set in (F q n , +).
Our construction needs many (not necessarily distinct) abelian skew Hadamard difference sets as input, where the exact number of skew Hadamard difference sets needed represents the "flexibility" of the construction. For example, we assume the existence of 25 skew Hadamard difference sets in abelian groups of order q to construct a skew Hadamard difference set in an abelian group of order q 5 . Thus, our construction seems to be very flexible. In fact, the construction can give rise to some inequivalent skew Hadamard difference sets when the group order is small. For example, Chen-Feng [3, Table 2] found by computer two inequivalent skew Hadamard difference sets D in (F 3 7 , +) with #Aut(Dev(D)) = 3 4 · 7 3 , where Aut(Dev(D)) stands for the full automorphism group of the design Dev(D). On the other hand, no general construction covering these examples have been known. Our construction covers these examples as explained below. Example 3. Let H i,0 , 1 i 6, be six skew Hadamard difference sets in (F q , +), and H i,1 , 1 i 6, be the inverses of H i,0 , 1 i 6, respectively. Furthermore, let D be the union of the following subsets of F 3 q : Then, our main theorem implies that D is a skew Hadamard difference set in (F 3 q , +). In particular, in the case where q = 7, the sets D with H 1,0 = H 2,0 = · · · = H 6,0 = {x : x is a nonzero square in F q } and H 1,0 = H 2,0 = · · · = H 6,0 = {x : x is a nonsquare in F q } give rise to two inequivalent difference sets in (F 3 q , +) with #Aut(Dev(D)) = 3 4 · 7 3 . Thus, this construction covers the aforementioned two inequivalent skew Hadamard difference sets in (F 3 7 , +).
In this paper, we give a recursive construction for skew Hadamard difference sets, which is a generalization of the construction in Example 3. We briefly explain the construction. Let n > 1 be an odd integer and G i , i = 0, 1, . . . , n − 1, be abelian (not necessarily elementary abelian) groups of order q. We assume that each G i contains some (not necessarily distinct) skew Hadamard difference sets H i . We construct a skew Hadamard i n − 1 as in Example 3. We will study how to choose such direct products in relation to an identity for coefficients of the Lucas polynomial and binomial coefficients. The main point of our recursive construction is that the assumed abelian groups containing skew Hadamard difference sets are not necessarily elementary abelian. Hence, if one finds a skew Hadamard difference set in a nonelementary abelian group, by plugging it into the construction, we obtain skew Hadamard difference sets in other nonelementary abelian groups. Therefore, we claim that if there exists a counterexample for Conjecture 2, then there exist infinitely many counterexamples for it.
The paper is organized as follows. In Section 2, we give an identity for coefficients of the Lucas polynomial and binomial coefficients, which is behind our construction. In particular, we need a constructive proof for the identity. In Section 3, we give our main construction for skew Hadamard difference sets based on the identity, and prove that the construction works well. In the final section, we apply our construction for small q's and discuss about the inequivalence of resulting skew Hadamard difference sets.

An identity for coefficients of Lucas polynomials and binomial coefficients
In this section, we study a relationship between coefficients of the Lucas polynomial and the binomial coefficients, which will be used to construct skew Hadamard difference sets. The Lucas polynomial L n (x) ∈ Z[x] of degree n is defined by the following recurrence relation: Remark 4. The following facts on Lucas polynomials are classically well-known. See, e.g., [11].
(1) L n (x) can be explicitly written as (2) Let b n,h , 0 h n, denote the coefficient of x h in L n (x). Then, b n,n−2k is the number of k-subsets without consecutive points of a set of n points on a circle.
The key part of this paper is to give a "constructive" proof for the following identity.
Proposition 5. For positive integers n and 0 k n, it holds that Proof. Let V = {0, 1, . . . , n − 1} be a set of n points on a circle in the ordering 0 → 1 → . . . → n − 1 → 0. Define S n,k to be the set of k-subsets without neighbors of V . Then, by Remark 4 (2), |S n,k | = b n,n−2k . For any S ∈ S n,k , let We now consider the following process to make a k-subset of V : choose 0 i k, S ∈ S n,k−i and T ∈ S + i . Then, we have (X :=)S ∪ T ∈ V k . Conversely, let X be any k-subset of V , and let S = S + = ∅. For all A = {a, a + 1, . . . , a + d} ⊆ X such that d 0 and a − 1, a + d + 1 ∈ X, we add a, a + 2, a + 4, . . . ∈ A into S and add their neighbors a+1, a+3, a+5, . . . into S + . In this way, we can determine unique 1 i k, S ∈ S n,k−i and T ∈ S + i such that X = S ∪ T ∈ T n,i (S). It is not difficult to see that this is the reverse process of the above. This implies that Then, which proves the proposition.
Remark 6. We should mention that the identity in Proposition 5 is a special case of the following more general formula: which is given in [9, Eq. (5.62)]. In fact, we obtain (1) by putting n, k, r, s, t in (3) as k, k − i, n, k, −1, respectively. This was pointed out by one of the reviewers. However, we could not find our constructive proof in the literature, and we particularly need the structural identity (2) in our construction for skew Hadamard difference sets.
• H S,i j ,1 ⊆ G i j , 1 j |R S |: the inverses of H S,i j ,0 , 1 j |R S |, respectively; the electronic journal of combinatorics 27(3) (2020), #3.36 • For each S ∈ S n,k , define D S as a union of all subsets A 0 × A 1 × · · · × A n−1 ⊆ G such that The following is our main theorem.
Then, D forms a skew Hadamard difference set in G.
Remark 9. We need n−1 2 k=0 |S n,k |(n − 2k) skew Hadamard difference sets in abelian groups of order q to define the set D. Note that by Remark 4 we have This number represents the "flexibility" of this construction of skew Hadamard difference sets.
Example 10. In the case where n = 5, the set D is the union of the sets in Table 2. We need 2

Collection of auxiliary lemmas
We now collect some auxiliary and elementary lemmas for proving Theorem 8.
Lemma 11. Let m = 2h + 1 be an odd positive integer, and let .
Since the coefficient of x m−2j y 2j in f (x, y) is equal to that in f (x, −y) for each 1 j h, implies that it is equal to 0. On the other hand, by f (x, y) = f (y, x), we conclude that the coefficient of y m−2j x 2j , 1 j h, in f (x, y) is also equal to 0. This completes the proof.
Lemma 14. For any positive integer , let One can prove this lemma similarly to Lemma 13. Hence, we omit the proof.

Proof of Theorem 8
In this subsection, we prove Theorem 8 by a series of lemmas. Proof. Recall that G = G 0 × G 1 × · · · × G n−1 . We first see that for any x ∈ G * , the set D contains either x or −x. Let I x ⊆ V be the set of zero-coordinates of x. By (2) in the proof of Proposition 5, we have V k = k 2 i=0 S∈S n,k−i T n,i (S). This implies that there exists a unique S ∈ S n,|Ix|−i such that I x ∈ T n,i (S). For this S, D S contains either x or −x.
This completes the proof of the lemma.
We next evaluate the character values of D. To do so, we first fix some notation: • G ⊥ i , 0 i n − 1: the character group of G i , 0 i n − 1, respectively; • For fixed ψ = (ψ 0 , ψ 1 , . . . , ψ n−1 ) ∈ G ⊥ , let I ψ ⊆ V be the set of indices i such that ψ i is trivial, and let J ψ = V \ I ψ .
• For E ⊆ G and A ⊆ V , let E |A denote the multi-subset of i∈A G i obtained from E by restricting its coordinates to A. For example, if n = 3, E = E 0 × E 1 × E 2 and A = {2}, the multiset E |A contains each element of E 2 exactly |E 0 | · |E 1 | times.
the electronic journal of combinatorics 27(3) (2020), #3.36 • For A ⊆ V , let ψ |A denote the character of i∈A G i obtained from ψ ∈ G ⊥ by restricting its coordinates to A.
We first evaluate the character values of D S , S ∈ S n,k .
Next, we treat the case where S + ∩ J ψ = ∅. Define Since Lemma 17. Let ψ = (ψ 0 , ψ 1 , . . . , ψ n−1 ) ∈ G ⊥ be nontrivial, and let S ∈ S n,k . If S + ∩J ψ = ∅ and ψ i is nontrivial for any i ∈ R S , it holds that Proof. By (11), we need to show that y∈E S ψ |R S (y) = −1± . By the definition of E S and Lemma 1, Then, by Lemma 11, we have This completes the proof of the lemma.
Proof. By (11), we need to show that y∈E S ψ |R S (y) = −(−q + 1) |R S |−t /2. We consider E S |R S ∩J ψ , which is the multi-subset of i∈R S ∩J ψ G i obtained from E S by restricting its coordinates to R S ∩ J ψ . The multiset E S |R S ∩J ψ contains every element of i∈R S ∩J ψ G * i exactly (q − 1) |R S |−t /2 times. Since x∈G * i ψ i (x) = −1 for any nontrivial ψ i , we have This completes the proof of the lemma.
We are now ready for proving Theorem 8.
Example 19. We consider the case where n = 3. In this case, we need six skew Hadamard difference sets in (F q , +) to apply Theorem 8. We choose either the Paley difference set or its inverse in F q as the six skew Hadamard difference sets in (F q , +). Then, there are 2 6 possible choices. In the cases where q = 7, 11, 19 and 23, we obtained exactly two inequivalent skew Hadamard difference sets D in (F 3 q , +) from the 2 6 candidates, both of which satisfy #Aut(Dev(D)) = 3 · ( q−1 2 ) 3 · q 3 . From Example 19, the following problem naturally arises.
Problem 20. If q > 3 is a prime and n = 3, does the construction give rise to at least two inequivalent skew Hadamard difference sets with #Aut(Dev(D)) = 3 · ( q−1 2 ) 3 · q 3 ? Example 21. We next consider the case where n = 5. In this case, we need 25 skew Hadamard difference sets in (F q , +) to apply Theorem 8. As in Example 19, we take either the Paley difference set or its inverse in F q as the 25 skew Hadamard difference sets in (F q , +). Then, there are 2 25 possible choices. In the cases where q = 3, we obtained exactly nine inequivalent skew Hadamard difference sets D in (F 5 q , +) from the 2 25 candidates, four of which satisfy #Aut(Dev(D)) = 5 · q 5 and five of which satisfy #Aut(Dev(D)) = q 5 .
The number of skew Hadamard difference sets as input for our construction increases according to the growth of n. Hence, the following question naturally arises.
Problem 22. We apply the recursive construction as G i = F q and H S,i,j either the Paley difference set in (F q , +) or its inverse. Does the number of inequivalent skew Hadamard difference sets in (F n q , +) obtained from the construction increase according to the growth of n?
The fact that we could have a recursive construction implies that there exist skew Hadamard difference sets in abundance. In fact, as already mentioned, if one finds a skew Hadamard difference set in a nonelementary abelian group, we obtain infinite families of skew Hadamard difference sets in nonelementary abelian groups by applying our recursive construction. Therefore, the author feels that Conjecture 2 is doubtful. Hence, we close this paper with the following important problem.
Problem 23. Can one find a skew Hadamard difference set in a nonelementary abelian group?