Designs, Codes and Cryptography, 17, 61–68 (1999)c© 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Williamson Matrices and a Conjecture of Ito’s

BERNHARD SCHMIDT schmidt@cco.caltech.eduDepartment of Mathematics, MC 253-37, Caltech, Pasadena, CA 91125

Dedicated to the memory of E. F. Assmus

Received June 9, 1998; Revised August 24, 1998; Accepted September 14, 1998

Abstract. We point out an interesting connection between Williamson matrices and relative difference sets innonabelian groups. As a consequence, we are able to show that there are relative(4t,2,4t,2t)-difference sets inthe dicyclic groupsQ8t = 〈a,b|a4t = b4 = 1,a2t = b2,b−1ab= a−1〉 for all t of the formt = 2a · 10b · 26c ·mwith a,b, c ≥ 0, m ≡ 1 (mod 2), whenever 2m− 1 or 4m− 1 is a prime power or there is a Williamson matrixoverZm. This gives further support to an important conjecture of Ito [11] which asserts that there are relative(4t,2,4t,2t)-difference sets inQ8t for every positive integert . We also give simpler alternative constructions forrelative(4t,2,4t,2t)-difference sets inQ8t for all t such that 2t−1 or 4t−1 is a prime power. Relative differencesets inQ8t with these parameters had previously been obtained by Ito [6]. Finally, we verify Ito’s conjecture forall t ≤ 46.

Keywords: Hadamard matrices, relative difference sets, Williamson matrices, Ito’s conjecture, dicyclic groups

1. Introduction

Let G be a group of orderm. An m× m matrix A is calledG-invariantif the rows andcolumns ofA = (ag,h) can be indexed with elementsg, h of G such thatagk,hk = ag,h forall g, h, k ∈ G.

A Hadamard matrix of orderv is av×v-matrix H with entries±1 satisfyingH Ht = v IvwhereIv is the identity matrix of sizev. It is well known thatv ≡ 0 (mod 4) if a Hadamardmatrix of orderv > 2 exists.

A Hadamard matrixH of order 4m is said to be a Williamson matrixover an abeliangroupG of orderm if H is of the form

H =

A B C D−B A −D C−C D A −B−D −C B A

(1)

whereA, B,C, D areG-invariantm× m matrices with entries±1. We remark that thisdefinition slightly differs from the usual one which requires thatXYt = Y Xt holds for all2-subsets{X,Y} of {A, B,C, D}. This property is stronger than the orthogonality of therows of H .

For the study of Williamson matrices and relative difference sets we will use the followinggroup ring notation. We will always identify a subsetA of a groupG with the element

62 SCHMIDT

∑g∈A g of the integral group ringZ[G]. For B = ∑g∈G bgg ∈ Z[G] we write B(−1) :=∑g∈G bgg−1. We may identify any elementS = ∑

g∈G sgg of the group ringZ[G] withtheG-invariant matrix(mg,h) wheremg,h = sgh−1. We note thatS(−1) corresponds to thematrix St . In terms of the group ring, necessary and sufficient conditions for a matrix ofthe form (1) to be a Hadamard matrix are

AA(−1) + B B(−1) + CC(−1) + DD(−1) = 4m (2)

and

XY(−1) − X(−1)Y + T Z(−1) − T (−1)Z = 0 (3)

for (X,Y, T, Z) = (A, B,C, D), (A, D, B,C), (A,C, D, B).Let G be a (possibly nonabelian) group of ordermn, and letN be a normal subgroup of

G of ordern. A k-subsetR of G is called an(m,n, k, λ)-differenceset inG relative toN

if every g ∈ G \ N has exactlyλ representationsg = r1r−12 with r1, r2 ∈ R, and no

nonidentity element ofN has such a representation. The following is a translation of thisdefinition into the group ring notation.

LEMMA 1.1 A k-subset R of a group G of order mn is a relative(m,n, k, λ)-difference setin G relative to a normal subgroup N of order n if and only if

RR(−1) = k+ λ(G− N)

in Z[G].

A relative(4t,2,4t,2t)-difference set is a special kind of semiregularrelative differenceset, see [2, 12]. Such a relative difference setR contains exactly one element of each cosetof the subgroupN. SinceN has order 2, we may identifyN with {−1,1}. Let g1, . . . , g4t

be a system of coset representatives ofN in G. Lethi j be the unique element ofRgi g−1j ∩N.

Then the definition of a relative difference sets implies that(hi j ) is a Hadamard matrix oforder 4t . These Hadamard matrices can also be obtained from 2-cocyles, see [5], and thusare sometimes called cocyclic. Ito [11] conjectured that relative(4t,2,4t,2t)-differencesets in the dicyclic groupsQ8t = 〈a,b | a4t = b4 = 1,a2t = b2,b−1ab= a−1〉 exist forall positive integerst . Ito’s conjecture is a very interesting strengthening of the outstandingHadamard conjecture which asserts that a Hadamard matrix of order 4t exists for everypositive integert . In [5], Ito’s conjecture was shown to be true fort ≤ 11 by a computersearch. We will verify Ito’s conjecture for allt ≤ 46.

2. The Connection

Let Q8 := 〈x, y | x4 = y4 = 1, x2 = y2, y−1xy = x−1〉 be the quaternion group of order8. We first show that a Williamson matrix over an abelian groupG of orderm is equivalentto a(4m,2,4m,2m)-relative difference set inG× Q8 relative to〈1〉 × 〈x2〉.

WILLIAMSON MATRICES 63

THEOREM2.1 A Williamson matrix over an abelian group G of order m exists if and only ifthere is a(4m,2,4m,2m)-relative difference set in T:= G×Q8 relative to N := 〈1〉×〈x2〉.Proof. Let R be a subset ofT containing exactly one element of each coset ofN. WriteU := G× 〈x2〉 andR= E+ Fx+ K y+ Lxy with E, F, K , L ⊂ U . ComputingRR(−1)

and using Lemma 1.1 we see thatR is a(4m,2,4m,2m)-relative difference set inT if andonly if

E E(−1) + F F (−1) + K K (−1) + LL(−1) = 4m+ 2m(U − N) (4)

E(−1)F + K (−1)L + (E F(−1) + K L(−1))x2 = 2mU (5)

F L (−1) + E(−1)K + (E K (−1) + F (−1)L)x2 = 2mU (6)

F (−1)K + E(−1)L + (E L(−1) + F K (−1))x2 = 2mU. (7)

Since each of the setsE, F, K , L contains exactly one element of each coset ofN in U ,we can writeX = X1 + X2x2 with X1, X2 ⊂ G andX1 + X2 = G for X = E, F, K , L.DefineA := E1− E2, B := F1− F2, C := K1− K2, D := L1− L2. It is straightforwardto verify that equations (4) through (7) hold if and only ifA, B,C, D satisfy (2) and (3)which proves Theorem 2.1.

For instance, assume thatA, B,C, D satisfy (2). We will show that this implies (4). Wehave to show

χ(E E(−1) + F F (−1) + K K (−1) + LL(−1)) = χ(4m+ 2m(U − N)) (8)

for all charactersχ of U . If χ if trivial on N = 〈x2〉, then (8) follows from the fact thatE, F, K , L contain exactly one element of each coset ofN in U . If χ is nontrivial onN, thenχ(x2) = −1 implying χ(E) = χ |G(A), χ(F) = χ |G(B), χ(K ) = χ |G(C),andχ(L) = χ |G(D). Thus (8) follows from (2) in this case. The proof of all remainingimplications is similar.

Let G be an abelian group of orderm. By Q(G) we denote the semidirect product ofGwith the quaternion groupQ8 = 〈x, y | x4 = y4 = 1, x2 = y2, y−1xy = x−1〉 which isgiven byG G Q(G), x−1gx = g andy−1gy = g−1 for all g ∈ G. Note that for oddm,the dicyclic groupQ8m := 〈a,b | a4m = b4 = 1,a2m = b2,b−1ab= a−1〉 coincides withQ(Zm). By the same arguments as in the proof of Theorem 2.1 we obtain the following.

THEOREM2.2 Let G be an abelian group of order m. A(4m,2,4m,2m)-difference set inQ(G) relative to〈y2〉 exists if and only if there is a Hadamard matrix of the form

A B C D−B A −D C−Ct Dt At −Bt

−Dt −Ct Bt At

(9)

where A, B,C, D are G-invariant m×m matrices with entries±1.

Note that a matrix of the form (9) is a Hadamard matrix if and only ifAA(−1)+ B B(−1)+CC(−1)+ DD(−1) = 4m andA(−1)B− AB(−1)+C(−1)D−C D(−1) = 0 in the group ring.

64 SCHMIDT

These conditions are weaker than (2) and (3) and thus we get the following result, a specialcase of which was obtained in [8, Prop. 3].

COROLLARY 2.3 The existence of a Williamson matrix over an abelian group G implies theexistence of(4m,2,4m,2m)-relative difference sets in G× Q8 and Q(G).

3. Relative Difference Sets in Q8m

We want to study relative(4m,2,4m,2m)-difference sets in dicyclic groupsQ8m = 〈a,b |a4m = b4 = 1,a2m = b2,b−1ab= a−1〉 in more detail. The following lemma is from [11].For the convenience of the reader, we include a proof.

LEMMA 3.1 A (4m,2,4m,2m)-difference set in Q8m relative to N := 〈b2〉 exists if andonly if there are polynomials f , g of degree2m− 1 and coefficients±1 only such that

f (x) f (x−1)+ g(x)g(x−1) ≡ 4m (modx2m + 1). (10)

Proof. Let R be a subset ofQ8m containing exactly one element of each coset ofN andwrite R= F + Gb with F,G ⊂ 〈a〉. ComputingRR(−1) and applying Lemma 1.1 showsthat R is a(4m,2,4m,2m)-difference set inQ8m relative toN if and only if

F F (−1) + GG(−1) = 4m+ 2m(〈a〉 − N). (11)

The equivalence of (10) and (11) can be verified by using the characters of〈a〉 similar tothe proof of Theorem 2.1.

We recall that a pair of Golay polynomialsis a pair of polynomialsf, g of degree 2m−1and coefficients±1 only such that

f (x) f (x−1)+ g(x)g(x−1) = 4m (12)

in Z[x, x−1]. Golay polynomials of degree 2m− 1 are known for allm such that 2m =2r ·10s ·26t with r, s, t ≥ 0, see [4], and it is conjectured that there are no Golay polynomialsfor any otherm. Of course, Lemma 3.1 shows that the existence of a pair of Golaypolynomials of degree 2m−1 implies the existence of a(4m,2,4m,2m)-relative differenceset inQ8m, but we can say more. The following construction is based on the observationthat an idea of Turyn [15, Lemma 5] still works in a slightly more general situation.

THEOREM3.2 If there is a pair k, l of Golay polynomials of degree2m− 1, and if thereis a (4m′,2,4m′,2m′)-relative difference set in Q8m′ relative to〈b2〉, then there is also an(8mm′,2,8mm′,4mm′)-difference set in Q16mm′ relative to〈b2〉.Proof. Let u, w be the polynomials corresponding to the relative difference set inQ8m′

via Lemma 3.1. Define

f (x) := 1

2

{u(x2m)[k(x)+ l (x)] + x4mm′−2mw(x−2m)[k(x)− l (x)]

},

g(x) := 1

2

{w(x2m)[k(x)+ l (x)] + x4mm′−2mu(x−2m)[−k(x)+ l (x)]

}.

WILLIAMSON MATRICES 65

It is easy to check thatf andg are polynomials of degree 4mm′ −1 with coefficients−1,1only. We compute

f (x) f (x−1)+ g(x)g(x−1) = 1

2[k(x)k(x−1)+ l (x)l (x−1)]

·[u(x2m)u(x−2m)+ w(x2m)w(x−2m)]

= 2m[u(x2m)u(x−2m)+ w(x2m)w(x−2m)]

≡ 8mm′ (modx4mm′ + 1)

sincek(x)k(x−1)+l (x)l (x−1) = 4mandu(x)u(x−1)+w(x)w(x−1) ≡ 4m′(modx2m′ +1).By Lemma 3.1,f andg describe the desired relative difference set.

Ito [6] constructed relative(4t,2,4t,2t)-difference sets inQ8t for all t such that 4t − 1is a prime power. This construction is of special interest since these relative difference setscannotbe derived from known families of Williamson matrices. It is not known whetherWilliamson matrices of order 4t , 4t −1 a prime power, exist in general. A computer search[3] showed that there is no Williamson matrix of order 4· 35 such that the four matricesA, B,C, D aresymmetric. However, it seems quite plausible that (without the symmetrycondition) Williamson matrices exist for all orders 4t even for generalt . In the following,we give a simpler alternative to Ito’s construction of relative(4t,2,4t,2t)-difference setsin Q8t for all t such that 4t − 1 is a prime power.

THEOREM3.3 ([6]) There is a relative(4t,2,4t,2t)-difference set in Q8t for all t such thatq := 4t − 1 is a prime power.

Proof. Let U be the subgroup of order(q − 1)/2 of F∗q2 whereFq2 is the finite field of

orderq2. Let tr denote the trace function ofFq2 relative toFq. We chooseα ∈ Fq2 withtr (α) = 0 and denote the set of nonzero squares inFq by Q. Then [12, Thm. 2.2.12]implies that

R := {U x: tr (αx) ∈ Q}is a relative(q+ 1,2,q, (q− 1)/2)-difference set inW := F∗q2/U . Let g be a generator ofW and write

R= R1+ R2g (13)

with Ri ⊂ 〈g2〉. Lemma 1.1 implies

R1R(−1)1 + R2R(−1)

2 = q + q − 1

2(〈g2〉 − 〈g4t 〉). (14)

If the multiplicative order ofd ∈ F∗q2 dividesq + 1, thentr (αd) = αd + αqdq = αd −αd−1 = −tr (αd−1) sincetr (α) = 0. ThusR(−1)

1 = g4t R1. Note 1 6∈ R1 sincetr (α) = 0.Thus

R1+ R(−1)1 = 〈g2〉 − 〈g4t 〉. (15)

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We identify the cyclic subgroup of order 4t of Q8t with 〈g2〉, i.e. we writeQ8t = 〈a,b |a4t = b4 = 1,a2t = b2,b−1ab= a−1〉with a = g2. DefineS := (R1+1)+R2b ∈ Z[Q8t ].Using Lemma 1.1, equations (14), (15), and the fact that bothR1+1 andR2 contain exactlyone element of each coset ofN := 〈a2t 〉 in 〈a〉, it is straightforward to verify thatS is a(4t,2,4t,2t)-difference set inQ8t relative toN.

REMARK 3.4 The arguments above can be used to give an interesting proof of the wellknown number theoretic fact that every prime powerq ≡ 3 (mod 8) is a sum of three oddsquares two of which are equal. To see this, we first note thatt = (q + 1)/4 is odd ifq ≡ 3 (mod 8). Thus we may writeR= R1 + R2gt with Ri ⊂ 〈g2〉 instead of (13). Fromthe proof of Theorem 3.3 we know

R(−1)1 = R1h (16)

whereh := g4t . In the same way, we obtain

(R2gt )(−1) = R2gt . (17)

Now we write R1 = X1 + X2g2t and R2 = X3 + X4g2t with Xi ⊂ 〈g4〉. From (16) and(17) we get

X(−1)1 = X1h,

X(−1)3 = X4h.

(18)

Equation (14) implies

4∑i=1

Xi X(−1)i = q + q − 1

2(〈g4〉 − 〈h〉). (19)

Let χ be a character of〈g〉 of order eight. Thenχ(h) = −1 andχ( f ) = χ( f −1) ∈{−1,1} for all f ∈ 〈g4〉. Thusχ(Xi ) = χ(X(−1)

i ) for all i . Moreover,χ(X1) = 0 andχ(X3) = −χ(X4) by (18). Note thatχ(Xi ) is odd fori = 2,3,4 sinceXi has exactlytelements fori = 2,3,4. Now (19) implies

q = χ(X2)2+ 2χ(X3)

2

which is the desired decomposition ofq into the sum of three odd squares.

Ito [7] obtained relative(4t,2,4t,2t)-difference sets inQ8t for all t such thatq :=2t − 1 ≡ 1 (mod 4) is a prime power. We give a simpler alternative construction whichalso works forq ≡ 3 (mod 4).

THEOREM3.5 There is a relative(4t,2,4t,2t)-difference set in Q8t for all t such thatq := 2t − 1 is a prime power.

Proof. Let G = 〈a〉 be cyclic of order 4t . By [12, Thm. 2.2.12] there is a(q+1,2,q, (q−1)/2)-difference setR in G relative toN := 〈a2t 〉. ReplacingRby Rgfor some appropriateg ∈ G if necessary, we may assumeR∩ N = ∅. We define

S := (R+ 1)+ (R+ a2t )b ∈ Z[Q8t ]

WILLIAMSON MATRICES 67

whereQ8t = 〈a,b | a4t = b4 = 1,a2t = b2,b−1ab = a−1〉. Using Lemma 1.1 it isstraightforward to verify thatS is the desired relative difference set.

Combining Corollary 2.3 and Theorems 3.2, 3.3, 3.5, we get the following result.

COROLLARY 3.6 Let m be a positive integer such that2m−1or 4m−1 is a prime power orm is odd and there is a Williamson matrix overZm. Then there is a relative(4t,2,4t,2t)-difference set in Q8t for every t of the form

t = 2a · 10b · 26c ·mwith a,b, c ≥ 0.

REMARK 3.7 Williamson matrices overZm are known for manym including allm of theform m = qr (q + 1)/2 whereq ≡ 1 (mod 4) is a prime power andr is any nonnegativeinteger, see [13, 14, 16]. Moreover, Williamson matrices overZm exist for allm≤ 33 andfor m= 39,43, see [1, 3].

In [5], a computer search was carried out which showed that(4t,2,4t,2t)-difference setsin Q8t exist for t ≤ 11. Combining Corollary 3.6 and Remark 3.7 we can improve thisresult considerably.

COROLLARY 3.8 There are relative(4t,2,4t,2t)-difference sets in Q8t for all t ≤ 46.

Acknowledgments

I would like to thank Thomas K¨olmel for several useful discussions and for convincing methat Ito’s conjecture is true.

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