on substructures of abelian difference sets with classical parameters

On Substructures of Abelian DifferenceSets with Classical Parameters∗

Kevin Jennings†

UCD School of Mathematical Sciences, University College, Dublin,E-mail: [email protected]

Received August 31, 2006; revised August 20, 2007

Published online 7 January 2008 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/jcd.20172

Abstract: We constrain the structure of difference sets with classical parameters in abeliangroups. These include the classical Singer [7] and Gordon et al. [4] constructions and also morerecent constructions due to Helleseth et al. [5], [6] arising from the study of sequences with idealautocorrelation properties. A unified overview of the known families is given in [2] and [3]. Weshow here that any abelian difference set with these parameters inherits a very regular intersec-tion property with regard to subgroups. We show in particular that a planar difference set can al-ways be found embedded in an abelian difference set of odd order whose parameters are those ofa 5-dimensional projective geometry. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 182–190, 2008

Keywords: difference sets; projective plane; abelian difference

1. INTRODUCTION

Let G be a finite abelian group of order v and let ZG denote the integral group ring of G.Given

∑aigi ∈ ZG, we set

a(−1) =∑

aig−1i .

Let D be a k-subset of G, where k ≥ 1. By a standard abuse of notation, we will use theletter D to represent both the set of elements D and the corresponding group ring elementD = ∑

d∈D d.We say that D is a (v, k, λ)-difference set in G if D satisfies the group ring equation

DD(−1) = λG + n1,

where n = k − λ is the order of the difference set.

∗This work is part of the author’s Ph.D. thesis.†Present address: Department of Mathematics, St. Patrick’s College of Education, Dublin, Ireland.

Journal of Combinatorial Designs© 2008 Wiley Periodicals, Inc.

182

ON ABELIAN DIFFERENCE SETS 183

We say that an automorphism σ of G is a multiplier for D if σ(D) = gD for some g ∈ G.More particularly, we say σ is a (numerical) multiplier for D if σ(x) = xm for all x ∈ G,where m is an integer relatively prime to v. We call gD a translate of the difference setD. Clearly gD is itself a difference set. Following [4], we say that the difference set D isnormalized if

∏d∈D

d = 1.

It is elementary to show that any difference set with gcd(v, k) = 1 has a unique translatewhich is normalized. It is straightforward to prove that such a normalized difference set isfixed set-wise by any numerical multiplier of D.

We recall the multiplier theorem of Marshall Hall and its various refinements which weadapt for our purposes below. For proofs see [1, p. 323].

Theorem 1 (M. Hall). Let D be an abelian (v, k, λ)-difference set where n = k − λ is apower of a prime p and gcd(p, v) = 1. Then the mapping σ : x → xp is a multiplier for D.

Suppose now that D is a (v, k, λ)-difference set whose parameters are given by

(v, k, λ) =(

qd − 1

q − 1,qd−1 − 1

q − 1,qd−2 − 1

q − 1

), (1)

where q is a power of a prime. Any difference set with these parameters is said to haveclassical parameters. Such difference sets include [4], [5], [6], [7] and a unified treatmentis given in [2], [3]. The order n = k − λ of these difference sets is qd−2 so Theorem 1 applies.

As it contains the key idea to our observation, we state here a condensed version of ageneralization of the Mann Test, the full statement of which, with proof, can be found in[1, p. 336].

Theorem 2 (Mann Test). Let D be a (v, k, λ)-difference set in an abelian group G of orderv. Let U be a subgroup of G and let G/U have exponent u∗. Suppose p is a prime notdividing u∗ and pf ≡ −1 mod u∗ for some f ∈ N. Then the following hold:

(a) n = p2jn′, where gcd(p, n′) = 1, for some j ∈ Z.(b) For all cosets Ug of U in G, the corresponding intersection numbers |D ∩ Ug| of D

relative to U are congruent modulo pj .(c) pj ≤ |U|.

Throughout this article, we assume that G is abelian but not necessarily cyclic. This isprobably not a significant generalization, as all known abelian difference sets with classicalparameters are cyclic.

Let D be a difference set with classical parameters (1). Suppose d = 2m is even. Then,we have

|G| = v = q2m − 1

q − 1= qm − 1

q − 1.(qm + 1).

Journal of Combinatorial Designs DOI 10.1002/jcd

184 JENNINGS

Let H denote a subgroup of G of size qm−1q−1 . We observe that H satisfies the role of U in the

Mann Test, with f = sm, where q = ps.

Proposition 3. Let D be a difference set in an abelian group G with classical parameterswhere d = 2m is even. Let H denote a subgroup of G of size qm−1

q−1 . Then the elements of Dare distributed through the cosets of H as

|D ∩ Hx| ={ |H | once,

qm−1−1q−1 qm times.

Proof. We note from the Mann Test that all intersection numbers of D relative to H arecongruent to each other modulo qm−1. Let si denote the intersection number |D ∩ Hxi|.Then si ≡ y mod qm−1 for some y ∈ Z where 0 ≤ y < qm−1.

From the basic relation

qm+1∑i=1

si = k,

we have

(qm + 1)y + qm−1r = q2m−1 − 1

q − 1(2)

for some r ∈ Z. Looking at (2) mod qm−1 we see that

y ≡ qm−1 − 1

q − 1mod qm−1

and so

y = qm−1 − 1

q − 1.

Substituting this value for y back into (2), we get

(qm + 1)qm−1 − 1

q − 1+ qm−1r = q2m−1 − 1

q − 1.

Multiplying this out yields r = 1.We deduce that there is only one possible distribution of the si. Exactly one coset has

intersection size

qm−1 + qm−1 − 1

q − 1= qm − 1

q − 1



while each of the qm other cosets has intersection size

qm−1 − 1

q − 1.

�

So, in particular, we have two possibilities for D ∩ H :

|D ∩ H | ={ |H | that is H ⊆ D or

qm−1−1q−1

and thus if H is not contained in D, then D ∩ H is the correct size to be a difference set inH with classical parameters (1) where d = m.

We can proceed to deduce further information about the structure of G and conclude thatH is unique.

Proposition 4. Let D be a difference set in an abelian group G with classical parameterswhere d ≡ 0 mod 4. If q is even, then the Sylow 2-subgroup of G is trivial. If q is odd, thenthe Sylow 2-subgroup of G is cyclic.

Proof. If q is even, then G has odd order and so the Sylow 2-subgroup is trivial. For therest of the proof, we take q to be odd.

We write d = 2m where m is even and we let q be a power of an odd prime. Then, asbefore

|G| = v = qm − 1

q − 1(qm + 1)

and we let H be a subgroup of order qm−1q−1 . We observe that |H | is even and that |G : H | is

even with 21 being the highest power of 2 dividing |G : H |. Let S be the Sylow 2-subgroupof G and suppose that S is not cyclic. We will show the existence of a subgroup N in Gwhich contradicts the conditions of the Mann Test.

Since S is not cyclic, there exists T ≤ S with

S/T ∼= Z2 × Z2.

Write

H = H0 × (H ∩ S)

so that H0 is the odd order part of H. Let

N = H0 × T,

so that G/N has Sylow 2-subgroup isomorphic to Z2 × Z2. Thus,

|G : N| = 2(qm + 1).


186 JENNINGS

Note that N satisfies the role of U in our statement of the Mann Test and that due to thenon-cyclic structure of G/N, the exponent u∗ of G/N is qm + 1 here. Hence, by the MannTest part (c),

qm−1 ≤ |N| = qm − 1

2(q − 1)

which is a contradiction. So S is cyclic. �

We remark here that if d ≡ 2 mod 4 and q ≡ 1 mod 4, then the Sylow 2-subgroup of G issimplyZ2. We cannot say anything in general about the case d ≡ 2 mod 4 and q ≡ 3 mod 4.

Lemma 5. Let D be a difference set in an abelian group G with classical parameters (1),where d = 2m is even. Then G has a unique subgroup H of order qm−1

q−1 .

Proof. As before, we have

|G| = qd − 1

q − 1= qm − 1

q − 1(qm + 1).

Note that gcd( qm−1

q−1 , qm + 1)

divides 2. We observe that if m is odd, then qm−1q−1 is also odd.

In this case,

G = HM

where M ≤ G with |M| = qm + 1 and H ∩ M = 1.If m is even, then Proposition 4 applies and the Sylow 2-subgroup of G is cyclic. This

ensures that H is unique. �

With the setup of Lemma 5, we can apply Theorem 1 to deduce that the mapping

σ : x → xq

is a multiplier for D. Let τ = σm. Then τ is an involution on G.

Lemma 6. With the above notation, the set of fixed points of τ is precisely the union ofall cosets of H of the form Hz, where z2 ∈ H .

Proof. Suppose x is fixed by τ. Then, we have

xqm−1 = xqm−1q−1 (q−1) = 1.

Since H is the unique subgroup of order qm−1q−1 , x must be of the form

x = hg

where h ∈ H, g ∈ G and gq−1 ∈ H . But gcd(q − 1, qm + 1) = 2 and hence g2 ∈ H . Clearlyany element of the form hz with h ∈ H and z2 ∈ H is fixed by τ and the result isproved. �



We see that τ acts as an involution on the cosets of H, sending the coset Hx to Hx−1. ByProposition 3, we have

|D ∩ Hx| =

qm−1q−1 for one distinguished coset,

qm−1−1q−1 for the other qm cosets.

The distinguished coset must be fixed by τ and hence, by Lemma 6, it must be of the formHz where z2 ∈ H . Note that the distinguished coset is contained inside the difference set.We observe that if q is a power of 2 then G has odd order and τ only fixes one coset whichmust be H itself. So the subgroup H is contained in the normalized difference set in thiscase. We generalize this result to the case where q is a power of an odd prime now.

Theorem 7. Let D be a normalized difference set with classical parameters in an abelian

group G where |G| = q2m−1q−1 and q is a power of an odd prime. Let H be the subgroup of G

of order qm−1q−1 . Then

(a) If m is odd, H ⊆ D.(b) If m is even, Hz ⊆ D, where z is a generator of the Sylow 2-subgroup of G.

Proof. We have that

D =⋃

(D ∩ Hx),

where the union is over all cosets of H in G. As we noted above, the action of τ on the cosetsof H is τ(Hx) = Hx−1.

We decompose the elements of D into those cosets Hx �= Hx−1 which are paired withHx−1 by τ, and those cosets Hz, where z2 ∈ H , which are fixed by τ.

The condition that D is normalized requires that

1 =∏d∈D

d =∏

d∈D∩Hz

d∏

d∈D∩Hx

d,

where z2 ∈ H and x2 /∈ H in the products above. Now∏

d∈Hx d ∈ H since τ(Hx) = Hx−1.Hence, we must have

∏d∈D∩Hz

d ∈ H,

where, as before, z2 ∈ H . Now for any z /∈ H with z2 ∈ H , the product

∏d∈D∩Hz

d = hz|D∩Hz|

for some h ∈ H . This product is in H if and only if |D ∩ Hz| is even.


188 JENNINGS

Now suppose m is odd. Then there is only one distinguished coset with |D ∩ Hz| = qm−1q−1 ,

an odd number, while the other coset(s) satisfy

|D ∩ Hz| = qm−1 − 1

q − 1,

which is an even number. So the product of each of these elements is in H. Hence, theproduct of the elements of the distinguished coset must also be in H. Since for this coset,

∏d∈Hz

d = hzqm−1q−1 = hz ∈ H

we conclude that z = 1 and thus H itself is the distinguished coset with |D ∩ H | = qm−1q−1 .

Suppose instead now that m is even. In this case, since the Sylow 2-subgroup is cyclic,there are only two candidates for the distinguished coset, H itself and Hz where z is agenerator of the Sylow 2-subgroup. Clearly the product of those elements in D ∩ H is in Hand hence we require the product of the elements in D ∩ Hz to be in H. Since qm−1

q−1 is evenin this case, while qm−1−1

q−1 is odd, we conclude

|D ∩ Hz| = qm − 1

q − 1.

�

It is tempting to speculate that the difference set property may be inherited by thesubgroup, H, if we allow the possibility that it be a trivial difference set. Unfortunately,it appears difficult to see whether or not we get a uniform number of copies of elements ofH expressed as differences from (D ∩ H) × (D ∩ H) in general. The data at our disposal,that is, the known difference sets with classical parameters, do not contradict this speculativeconjecture but these data are based on methods from finite fields so we must be wary. Thereis a particular case where we can establish the existence of planar difference sets in theintersection, corresponding to parameters (1) where d = 6.

For the following result, we require knowledge about the Sylow 2-subgroup of G. Observethat if q ≡ 1 mod 4 then the Sylow 2-subgroup is simply Z2.

Theorem 8. Let D be a normalized difference set with classical parameters in an abelian

group G where |G| = q2m−1q−1 and m is odd and q is a power of an odd prime. Let H be the

subgroup of G of order qm−1q−1 . If Syl2(G) is cyclic, with z being the unique element of order

2 in G, then each element h ∈ H can be expressed an odd number of ways as

h = ab−1

where a, b ∈ D ∩ Hz. In particular, if m = 3, then the translate Dz ∩ H is a planardifference set in H.



Proof. We give a parity argument. Observe that λ = q2m−2−1q−1 is an even number. As before,

let τ be the involutary multiplier τ : x → xqm. Let h ∈ H . Then if

h = ab−1

with (a, b) ∈ D × D, we have

τ(h) = h = τ(a)τ(b)−1

where (τ(a), τ(b)) ∈ D × D. So difference representations of h come in pairs from thoseelements of D which are not fixed by τ, giving a total number which is even. Since weassume that Syl2(G) is cyclic, the subgroup of fixed points of τ consists of H and its cosetHz, by Lemma 6. Appealing now to Theorem 7, we see that H is contained in the differenceset and thus gives qm−1

q−1 representations of h, which is an odd number. Hence, h must arisean odd number of times as a difference representation from the other fixed coset, D ∩ Hz.

In particular, if m = 3, then |D ∩ Hz| = q + 1 and we conclude, by counting, that eachelement of H has exactly one difference representation from the intersection D ∩ Hz.

We see that D ∩ Hz is a translate of a planar difference set and hence the (not normalized)difference set Dz has the property that Dz ∩ H is a (normalized) planar difference setin H. �

2. DISCUSSION

We can discuss the possibility of an abelian difference set with classical parameters (1)having a non-cyclic Sylow 2-subgroup. If d is odd then G has odd order and Syl2(G) istrivial. If d ≡ 0 mod 4 then Proposition 4 assures us that Syl2(G) is cyclic. If d ≡ 2 mod 4and q is even or q ≡ 1 mod 4 then Syl2(G) is trivial or just Z2. So the only unresolvedsituation is when d ≡ 2 mod 4 and q ≡ 3 mod 4. The first such case is when d = 6 andq = 3, corresponding to a (364, 121, 40)-difference set. By an exhaustive computer search,Raja Mukherji (UCD) and I have been able to show the nonexistence of such an abeliandifference set with non-cyclic Sylow 2-subgroup. The restrictions imposed by Proposition 3on the cosets of H were crucial in reducing the number of cases to be considered. However,this technique alone is not strong enough to deal with larger values of q. This situationremains unresolved.

It is interesting to note that Theorem 8 guarantees an embedded planar difference setin the GMW difference sets with parameters of a 5-dimensional projective geometry. Itis, however, apparent that this embedded planar difference set is equivalent to a Singerdifference set, as we sketch below.

Let F be a field of order q and let K be the extension field of order q6. Let M and N beintermediate fields of orders q3 and q2, respectively. Let π(D) be a classic GMW differenceset in K∗/F∗, where π is the natural projection of scalars and D ⊆ K∗. Then D can beconstructed in essentially two ways in K∗ and we show that in either case, the elements inπ(M∗) have a Singer structure.

Since K∗ is cyclic, we are interested in the elements of D in M∗, since these are theelements projected into the subgroup of order q2 + q + 1 in K∗/F∗. As a first case, D canbe constructed as a union of M-hyperplanes in K, indexed by a K-hyperplane in M. This


190 JENNINGS

K-hyperplane in M corresponds to a Singer structure in π(M∗). Alternatively, D can beconstructed as a union of N-hyperplanes in K, indexed by a K-hyperplane in N (actually aline in this instance). But then these N-hyperplanes in K project onto a Singer differenceset of order q2 in the subgroup of K∗/F∗ of order q4 + q2 + 1. This Singer difference setis well known to contain a Singer subdifference set of order q in the subgroup of K∗/F∗ oforder q2 + q + 1. In either case, the classical GMW construction with d = 6 gives rise toan embedded Singer planar difference set.

It could be an interesting topic for further research to investigate the new and emergingfamilies of difference sets with parameters (1) such as the HG [5] difference sets and toverify that the embedded planar difference set when d = 6 and q is odd is indeed a Singerdifference set.

ACKNOWLEDGMENTS

The author is very grateful to Rod Gow (UCD) for his guidance and advice. The authorthanks Dieter Jungnickel and the referees for some helpful suggestions and Raja Mukherji(UCD) for assistance with the computer search.

REFERENCES

[1] T. Beth, D. Jungnickel, and H. Lenz, Design theory, Vol. I, Second Edition, Encyclopedia ofMathematics and its Applications, Vol. 69, Cambridge University Press, Cambridge, 1999. MRMR1729456 (2000h:05019).

[2] J. F. Dillon and H. Dobbertin, Cyclic difference sets with Singer parameters (odd q), preprint.

[3] J. F. Dillon and H. Dobbertin, New cyclic difference sets with Singer parameters, Finite FieldsAppl 10(3) (2004), 342–389. MR MR2067603 (2005f:05024).

[4] B. Gordon, W. H. Mills, and L. R. Welch, Some new difference sets, Canad J Math 14 (1962),614–625. MR MR0146135 (26 #3661).

[5] T. Helleseth and G. Gong, New nonbinary sequences with ideal two-level autocorrelation, IEEETrans Inform Theory 48(11) (2002), 2868–2872. MR MR1945579 (2003m:94055).

[6] T. Helleseth, P. V. Kumar, and H. Martinsen, A new family of ternary sequences with ideal two-level autocorrelation function, Des Codes Cryptogr 23(2) (2001), 157–166. MR MR1830946(2002e:94090).

[7] J. Singer, A theorem in finite projective geometry and some applications to number theory, TransAmer Math Soc 43(3) (1938), 377–385. MR MR1501951.


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