On the non-existence of generalised Hadamard matrices

Journal

of Statistical

Planning

and Inference

10 (1984) 385-396

385

North-Holland

ON THE NON-EXISTENCE MATRICES Warwick

de LAUNEY

Department

of Applied

Received

Abstract;

by Jaya

C,,,

to prove

g=O

Key

Subject words:

matrices;

of a generalized is shown

the non-existence

are divisible

N.S. W. 2006, Australia

of many

Generalised

matrix number

generalised

over its group

theoretic

Hadamard

the GH(lS,Crs),

ring factored

properties. matrices

GH(lS,Cs)

n and G we find the set of determinants

Classification:

Differences

Hadamard

to have certain

by 3, 5 or 7. For example

not exist. Also for certain AMS

of Sydney,

Srivastava

The determinant

by the relation orders

University

HADAMARD

5 May 1983

Recommended

ploited

Mathematics,

OF GENERALISED

out

These are ex-

for groups

whose

and GH(15,c~)

of the GH(n,G)

do

matrices.

05B99, 05B15, 05B20. Bhaskar

matrices;

Rao

Signing

designs; BIBD’s

Generalised over groups.

weighing

matrices;

Hadamard

1. Introduction Generalised Hadamard matrices are related arrays. They have also been used to construct of mutually orthogonal F-squares (Seberry, A generalised Hadamard matrix GH(n, G) matrix [ijj] whose entries are taken from G

to finite geometrices and orthogonal block designs (Seberry, 1979) and sets 1980). for a group G of order n is an n x n such that

whenever i#j and the summation is in the group ring. A number of authors have studied these matrices (Drake, 1979), (Jungnickle, 1979), (Shrikhande, 1964), (Street, 1979) and (Seberry, 1979, 1980). For a summary of the known matrices see Theorem A (Street, 1979). Also see note added in proof on last page. In this paper we develop number theoretic implications which are used to give non-existence theorems for GH(n, G) matrices when n is odd and 3, 5 or 7 ( IG(. Arising naturally from this approach is a method of finding the set of values taken by the determinants of the generalised Hadamard matrices of some orders for the groups C’s and C,. 0378-3758/84/$3.00

0

1984, Elsevier

Science

Publishers

B.V. (North-Holland)

W. de Launey / Existence of generalised Hadamard

386

2. Definitions

and a preliminary

matrices

result

Let G be an abelian group, S(G) be the group ring of G, and R(G) be the ring of elements of the form c agg

where a,eQ,

EeC

with the usual polynomial

addition and the following multiplication:

We see S(G) is a subring of R(G). g)S(G) and let Q(G) =R(G)/(C,,, g)R(G). We see Let Z(G) = S(G)/(C,,, Z(G) is a subring of Q(G), and that Q(G) is a (/G I- 1)-dimensional vector space over Q with basis {g(gEG,g#l}. Now if H is a GH(n, G) matrix and H* is the matrix obtained from H by transposing H and inverting each entry then HH* = nI, where the matrix multiplication identity matrix. It follows that

is over Z(G) or Q(G) and Z,, is the n-dimensional

det(H) det(H*) = n” and hence that det(H) Q(G) and hence

is invertible in Q(G). So n-‘H*

is the inverse of H over

(H*)H = nl,, over Q(G) and Z(G). Proposition

2.1. If H is GH(n, G) then HT is GH(n, G).

Proof. From the last relation H* is a GH(n, G). But G is abelian and so inversion 0 is an isomorphism of G. It follows that HT is GH(n,G).

We now restrict our attention to the case G = CT,,,= (a 1am = l), mr2. case det(H) is a polynomial in a. Let m(a)=det H, then det(H*)=m(a-‘). So we have:

In this

Proposition 2.2. Zf there exists a GH(n, C,) then there exists an element m(a) E Z(G) such that m(a)m(a-‘) = n”.

W. de Launey / Existence of generalised Hadamard matrices

3. Number

theoretic

We reinterpret

properties

Proposition

387

of the determinant

2.2 to give restrictions

on the values

of n for which

there exists a GH(n, C,). Q(C,,,) can be thought of as a vector space over Q of dimension m - 1 with basis as a subring of Q(C,). Let us consider an immediate {l,a, . . ..o~-‘} and Z(C,) restriction placed by Proposition 2.2 on n in the case m = 3. Here m(a) = A + ,ua some A, fi E Z and we require (A +pu)(A We can rewrite

or

+,~ua-‘) = n”

A2+p2-jfA

= n”.

this as

4n” = (2A -&

+ 3$.

So n” must be representable as (Y’+ 3p2, a, PEZ. If we can write n as 3’m, m = 5 (6) then n” cannot be so represented. Hence no GH(n, C,) exists for n = 3’m, m = 5 (6). We can develop more restrictive conditions. Let m-2

m-2 t =

c

and

A;a’

s = c

i=O

where

,~~a’

i=o

Ai, ,ui E Z. Now define m-2

These

maps have a number

of properties.

Lemma 3.1. 1. (i) [st]=[s]o[t]. (ii) [s + t] = [s] + [t]. (iii) [t] is a linear map. 2. (i) INV is a ring homomorphism. Proposition

3.1.

If there exists a GH(n, C,,,) then there exist integers &,,A, ...,Am_2

such that

=

Proof.

n”Z,_t

= [n”] = [m(a)m(a-‘)I

[m@-‘)](t) = m(a-‘)t

= [m(a)] 0 [m(a-‘)I.

#I,_,.

Now

= (INV m(a))t = INV(m(a)INV(t))

388

W. de Launey / Existence of generalised Hadamard =

INV([m(a)](INV(t)))

matrices

= INV [m(a)] INV(t).

So n”l,,, _ 1= [da)] INV [m(a)] INV. NOW supose ~(u)=~O+~~U+...+~,_~U~-~,

LiEZ.

Then

[m(u)] = (m(u), m(u)u, m(u)u2, . . . , m(u)um-2) where [m(a)] is written as a matrix with respect to the basis {1, a, u2, . . . , umw2}. Thus Am-2-Am-3

*a*

A2-A,

Now

INV(l) = 1,

INV(+-~-Q-...-~~-~,

INV(u2) = F2,

... ,

INV(U~-~) = u2,

so

1 -1 o-1 INV = . . 0 -1

0 0 0 i 1

...

0 0 1

...

0

0

0 *** 0

Hence

We now have another form of Proposition existence theorems. Proposition

3.1 which allows us to obtain the non-

3.2. If there exists a GH(n, C,,,) then there exist integers vo, vl, . . . , v, _ I

such that v~+v~+v2+“‘+v,_~

=o,

v~+v~+v$+..~+v~_~

=2n”,

(vg + v#+

(v, + v,)2+ ~~~+(~,_~+~~)~=2rz”,

(VO+Vl+V2)2+(V*+V2+V3)2+***+(Vm_1+VO+V1)2=2nn,

W. de Launey / Existence of generaiised Hadamard matrices

Proof.

389

In the matrix [m(a)] INV we dot the first column with the first row to obtain 1~+~:+...+L~_,-(~,~,+1,I,+...+~,_,~,_,)=n”.

We dot the first column with the (i+ 1)th row to obtain (lo,l,,...,

~m-2)‘(0,~0,~,,...,~m-3)

= (lo,;l,,..., for lsirm-2.

Am-2)*(Ai,A;+i, ***tJm-2,0,&9J1, ***,Ai--Z)

So A;+A:+ **+& . ..r.lj-2) = ll” -(&,A1, . . . . Am-*)* (li,JLj+i, ~~~~~2,_~,O~~O,~i,

for 15 i 5 m - 2. The only subscripts which occur once in the dot product are i - 1 and m - 1- i. The rest occur twice. So multiplying by 2 and grouping according to a2 + 2ab + b* = (a + b)* gives ~f~~+~~~~~~+(~~~~~)~+(~j+~~A~)~+~~~+(~~~*~~~~*~~)~ +(~,-1,_i)*+‘.. for lsirm-2. v,_, =-I,,_*.

+ (XrLi_*-Lm_*)*= 2n”

Set vo=Ao, vl=A1-Lo, v~=~~-~,,...,v,,,_~=&,-~-&,-~, Now the ith row of [m(a)]INV is

and

(IZi_~,~i-~O,IZi+~-IZ~,...,IZm_*-~m_*-i, -~m-*-i,JLO-~m_i9

***,Li_s-Am_s)

for 15 i I m - 1. So we see that the entries of [m(a)] INV are quite special. It can also be checked that this row is (VO+V,+“.+Vi_],V]+V*+“‘+Vi,...rV,_*+V,_1~VO+”’+Vi-3)~

Noting that

Ai_2-A,_2=V,_1+Vo+V1+.‘.+Vi-2

we

see

(V~+V~+“~+Vi_~)*+**~+(V~_*+V~+V~+~~~+Vi_*)*=2?2”

for lsirm-1.

From the definition vo+vi+v,+

of Vi, 05ilm-1,

-**+ v,_i = 0.

So each equation in the above set is repeated. lriI:+(m-1). 0

4. Non-existence

theorems

When m = 3 Proposition v,+v,+v*=o,

we see

A complete

set is given when

for the cases m = 3, 5 and 7

3.2 requires that there exist integers vo, vi, v2 such that v;+v:+v:=2nn.

W. de Launey / Existence of generalised Hadamard matrices

390

We have: Theorem

4.1. Let

n = 3’pflp? ...pF, n odd. Then no generalised Hadamard matrix for C3 of order n exists if for some pi = 5 (6), ki is odd. Proof.

We prove that if there exist integers mO,ml such that mi+ (-ml)* + (ml - mo)* = 2k

and that if p 1k, p = 5 (6), then p* 1k. This is done by showing that if the residues vo, vi, v2 (mod p) satisfy v()+vr+v*=o

(p),

v;+v:+v:=o

(p),

then vc, vi, v* = 0 (PI. We may suppose ~~$0 (p); then we have l+(v,vgr)+(v*vgr)=o

(p),

Letx~v,v~‘.Thenwesee1+x2+(1+x)*= has a solution., Equivalently, 4x2+4x+1 So ($)

= 1-4(p)

I*+ (v,v~y+(v2v~1)* -O(p),

or

(2x+1)*=-3

and hencethat

= 0 (p). 1+x+x2=0(p)

(p).

= 1 (p), where (g) is the Legendre symbol. Noting -1 ( 7= >

1, P=l L-1, p=3

(4), (4),

we find -3

(

-= P

>

1, ~‘1 -1, p=-1 L

(61,

(4),

It follows that for p=5 (6), vo, vl, v2=0 (p). So our initial equation reduces to P;+(-Pr)*+(Pr-Po)*

= 2/P,

Prl,

whenever the highest power of p dividing k is odd. But this forces p* 1lp, giving a contradiction. 0 As yet there is no general result for any value of m # 3. One can, however, easily derive some powerful results for C5 and C,. We show that if there exist integers Ao,A,, A2,As such that

391

W. de Launey / Existence of generalised Hadamard matrices

then p 1k implies p 1&,A,, 22, A3 for p = 3,7. If we assume this is true then we have the following

theorem:

4.2. If n = 3kl 7k2 m, 3 {rn and 7 7 m where m is odd and one of k, or k, is odd then no GH(n, C,) exists.

Theorem

Proof.

By Proposition

3.1 there exist lo, &,A2,,13 such that

= k14

where k=2(3k17kzm)“=23’17’2r

some t,, t2,r. Now 3{r and 7{r. loss of generality we may assume t, is odd. we have po, pl, p2, p3 such that

or t2 is odd. Without By our assumption

Also one of tl

2

! some

PO

Fl-pO

L(2-pl

Pl

PI--PO

P3-PI

P2

P3--PO

P3

-

-P2

--PI

=

3rZ4,

PO-P2

PO--PI

PO

r not divisible

contradiction.

p3--r(l2

1

Pul -P2

by 3. But by our

assumption

this implied

32 13r, giving

a

0

We now prove that assumption. in question requires the existence

From the proof of Proposition 3.2 the equation of integers vo, vi, v2, v3, v4 such that

vO+vi+v2+vj+vz,=O,

(i)

(4.1)

v;+v:+v;+v;+v;=2k,

(ii)

(v. + Q2 + (v, + v2)2 + (v2 + v3)2 + (v3 + v~)~ + (v4 + vo)2 = 2k.

(iii)

Remark 4.1. We note that if (vo, vi, v2, v3, v4) is a solution then so are (vi, v2, v3, v4, v,), (1, v,vo’, v&, v34, vqvi’) and (v4, v3, v2, vl, vol. If 3 1k we have that {vi, VT,v:, v:, vi} is either (1, 1, l,O,O} or (O,O, O,O,O} (mod 3). so (v& v:, v$ v:, vi) = (l,O, l,O, 1) or (1, 1, l,O,O) (mod 3). Since (i) must hold, (~0,

Neither

we have without

~1, ~2, ~3, ~4)

of these satisfy

3(k, then 3

f

(40,

loss of generality

LO,

1) or

(1,L

LO,

0)

(mod

3).

(iii), so {v& v:, vi, v$ vi} = {O,O, O,O, 0} (mod 3). Hence

\AO,&,A2,A3&

When p = 7 we have four distinct

cases:

if

392

W. de Launey / Existence

of generalised

Hadamard

matrices

{1,2,2,2,0),

{v~,v:,v~,v~,v~}E{~,~,~,~,o),

(2,1,1,1,2}

or (O,O,O,O,O} (mod7).

The first case gives 2 possibilities: (1,2, -3,O,O) and (1,0,2, -3,0). The partition is made according to the positions of the non-zero entries. None of these satisfy (iii). No vector (vO,vlr v2, v3, vq) corresponding to the second case satisfies (ii). The third case can be partitioned: (3,1, -1,1,3), (3, -1,3,1, l), (3, -1, l,l, 3) or (3, -1,1,3,1). None of these satisfy (iii). Thus {vg, vf, v:, vi, vi} = {O,O,O,O,0) (mod 7). Thus our assumption is true and our theorem holds. When m = 7 Proposition 3.2 requires that there exist integers vo, vr, v2, v3, v4, vs, v6 such that (i) (ii) (iii) (iv)

vO+v~+v2+v3+v4+v5+v6=o,

v;+v~+V;+v;+v:+v:+v;=2k, (vr +‘v,)2+[v, +v2)2+(v2+v$+(vs+ + (V4 + Vs)2 + (Vj + V6)2 + (V, + VO)~ (vg + VI + v2)2 + (VI + v2 + v3)2 +

(V,

-k V5 +

V6)2

+ (Vg

+ (v2

+ V6 +

VO)~

+ t

=

v/$)2 2k,

v3 + (V,

v4)2

+ (v3

+ VO + VI)~

+

v4 +

=

2k.

v5)2

If 3 ( k then without loss of generality we have (vo, vi, v2, V3,

V4,

V5,

V6)

=

(1,

1, l,o,o,

(1,

LO,

(l,l,l,LLLO),

0, o), 1,40,0),

(1,0, 1,0, (19

1,0,

o),

1,40,1,40),

(Q494440)

(mod3).

Of the first five only the second satisfies (i), (ii) and (iii). The second, however, does not satisfy (iv). This demonstrates the effect of the fourth condition. Irrespective of this we have that if 3 1k then 3 ) v 0s v 1, v 29 v 39 v 49 v 5, v 6. Hence we have: Theorem 4.3. If n =3klm, exists.

3{m,

where m and k, are odd then no GH(n, C,)

wh ere G is abelian then there exists an onto homomorphism Now ifpjIG[ 4 : G + C, so we may apply Lemma 1.1 (de Launey, 1983). Hence we may group Theorems 4.1, 4.2 and 4.3 to obtain: Theorem 4.4. If )G ) is divisible by 3, 5 or 7 then there is no GH(n, G) when one of the following occurs: andki oddforsomepi=5(6). (i) 3 ( n and n =p:lp$ *e-p?, pif2, for lrist, (ii) 5 ( n and n = 3kl 7k2m, 3 { m and 7 {m, where m and one of k, and k2 are odd. (iii) 7 1n and n = 3klm, 3 {m,

where m and k, are odd.--

W. de Launey

/ Existence

of generalised

Hadamard

393

matrices

It will be noted by the reader that all our results preclude the appearance of some odd primes as odd powers in the prime decomposition of odd orders. The effect of the even prime, 2, appears to be quite different and the reader is referred to Theorem A (Street, 1979) where it appears as both even and odd powers in the orders of generalized Hadamard matrices for arbitrary Cp. As applications of Theorem 4.4 we have that the GH(15,C3) (by (i)) and the GH(15,Cts) does not exist. This does away with the possibility of using a GH(l5, Ci5) to construct a complete set of 15 x 15 orthogonal latin squares.

5. On the determinant

We find, for certain given n and G, the range of values taken by the determinants of the GH(n, G) matrices. We do this by solving for [m(a)] INV in the matrix equation of Proposition 3.1. Although it is not in general certain that every solution corresponds to the determinant of a generalised Hadamard matrix, in the cases we consider we can be sure that this is the case because of the essential uniqueness of the solution. We first prove the following statement is true for p = 3,5. If there exists &,,A,, ...,Ap_2 such that

= kZP-l

wherep21k, thenpIlo,A, ,..., Lp_2. For p= 3 all we need prove is that,(vo, vt, v2) = (O,O,0). If 32 1k and 3 7 vo, vI, vz then without loss of generality (vo, vi, v2) = (1, 1,l). So there exist ko, kl, k2 such that

(1 +3ko)2+(1 +3k1)2+(l

+3k2)2=

0 (9),

(1 + 3ko) + (I+ 3k,) + (1 + 3k2) E 0 (9). But this gives 6 = 0 (9) which is a contradiction. Now we prove the statement for p= 5. The quadratic and -1. So (V~,V:,v~,V:,v~}~{O,0,0,0,0}, (-l,-l,O,O,O}

{1,1,1,1,1},

residues (mod 5) are 1 (-1,-1,-1,-1,-l},

or {l,-l,l,-LO}

(mod5)

Our aim is to show {vf, v:, vf, v:, vi} = {O,O,O,O,0) (mod 5). By application of Remark 4.1 the second and third cases are equivalent. The fourth possibility is removed because equation 4.1(i) cannot be satisfied. In the second case we have without loss of generality vo, vi, v2, v3, vq= 1 (mod 5). So there exists integers ko, k,, k2, k3, k4 such that

W. de Launey / Existence of generalised Hadamard matrices

394

(1+5k0)*+(1+5k,)*+~~~+(1+5k4)*=0

(mod25),

(1 + 5k,,) + (1 + 5kr) + -.. + (1 + 5k,) = 0 (mod 25).

These equations give 3 = 0 (mod 25) which is a contradiction. So if 5 { li for some i = 0,1,2,3 then the last case holds and without loss of generality {v,,, vlr v2, vs, v4} = {1,2, -1, -2,O) (mod 5). Now either the sequence 1,l will occur in (vf, vf, v$, v$ vi), the sequence -1, -1 will occur or neither. The three possible cases are Case 1:

(vi, v:, v:, vf, vi) = (1, LO, -1,l)

(5),

Case 2:

= (1, 1, -LO, -1) (5),

Case 3:

= (1, -1, l,O, -1) (5).

Without loss of generality we may begin with vo= 1 (5). We first note that in cases 1 and 3 (v. + vi)* + (vi + v*)~+ (~2 + vs)* + (vs + v.,)* + (v4 + vo) = + 1 (5). So cases 1 and 3 are ruled out. We now deal with the case 2. If (vo,vl, v2, v3,v4) is a solution (mod 25) then (v. + 5k, v1 + 5k, v2 + 5k, v3 + 5k, v4 + 5k) is also. So we may assume v3 E 0 (25). This requires the existence of integers ko, k,, k2, k3 such that (1 + 5ko)* + (-1 + 5k,)* + (1 + 5(k, + k2))*+

(2 + 5k2)* + (-2 + 5k3)* = 0 (25),

(2+ 5k,)*+ (-2+ 5k,)*+ (-1 + 5(k,+ ko))* = 0 (25).

This means 10+lO(ko-kl)+20(k2-k,)=O

(29,

lO+ lO(ki - ko) + 30(k2 - k,) = 0 (25), which implies 20 = 0 (25) giving a contradiction. It follows that (vi, VT,v$, v:, vi> 5 (O,O,O,O, 0) and hence 5 ) lo, Al, AZ, A3. These results allow us to find the determinants of large classes of generalised Hadamard matrices for the groups C3 and Cs. Example

5.1. Let H be GH(3m,C3). Then

det(H) =

even,

+a’(1 + a)3rn’*,

m

&af(2+a)3m-“2,

m odd,

where TV {0,1,2}. We may deduce this in the following manner. We first note our matrix equation becomes

where k=

1 3

and

Ai =

3+mPi,

m even,

3’(m-‘)pi,

m odd.

W. de Launey / Existence of generalised Hadamard matrices

But the representations

of k= 1 and 3 are unique:

12+(-1)2+0=2*1 22+(-1)2+(-1)2= Noting

395

for k= 1, 2.3

for k=3.

m. = vo, ml = v. + v1 we have the solutions

[m(a)] INV=

We note multiplying by the factor -a’ corresponds to multiplying a column or row of H by a’ and switching a pair of rows or columns. It will be seen that permuting vo, vt, v2 and changing their signs will also produce the complete set of solutions. Similarly we have for H, GH(Sm, C,), that det(H)

=

+a’(1 +a+a2+a3)5m’2,

m even,

+ - a’(1 _ 2a2 _ 2a3)5(m-IV2

m odd.

,

where i E {0, 1,2,3,4}. Here 1

0

0

0

m even, 0

l-1

0

[m(a)] INV= l-l

2

0 ,

m odd.

It seems certain that non-existence theorems for generalised Hadamard matrices for C,,p any prime, are contained in Proposition 3.1, and that in their most general form they would take the form of Theorem 4.1. One could obtain particular instances of these, such as Theorem 4.1 and 4.3, by the use of a computer to deal with all the cases but this is unsatisfactory and what is called for is a complete solution of the set of equations in Proposition 3.2. The author is currently working on this problem.

396

W. de Launey

/ Existence

of generalised

Hadamard

matrices

Note added in proof

Dawson (1984) has proved GH(4q,EA(q))

exists for q a prime power.

Acknowledgement

The author wishes to thank his supervisor Dr. Jennifer Seberry for her help and encouragement.

References Shrikhande, J. Math. Jungnickel,

S.S. (1964). Generalised

Hadamard

matrices

and orthogonal

arrays

of strength

2. Canad.

16, 736-740. D. (1979). On difference

matrices,

matrices. Math. 2. 167, 49-60. Drake, D.A. (1979). Partial-geometries

resolvable

transversal

and generalised

Hadamard

designs

and generalised

matrices

Hadamard

over groups.

Canad.

J.

Math 31, 617-627. Seberry,

J. (1979).

Some remarks

SBIBDs. Combinatorial Berlin-Heidelberg-New Street,

on generalised

Mathematics York, 154-164.

D. (1979). Generalised

Hadamard

Hadamard

VI,

Lecture

matrices

Notes

in

and theorems Mathematics

matrices,

orthogonal

arrays

for generalised

Hadamard

matrices.

of Rajkundlia Vol. 748.

and F-squares.

on

Springer,

Ars Combinatoria

8, 131-141. Seberry,

J. (1980). A construction

365-368. de Launey, binatorial

W. (1983).

Generalised

Mafhematics

New York. Dawson, J. (1984). A construction Inference

11, to appear.

Hadamard

X, Lecture

Notes

matrices

whose

in Mathematics,

for generalized

Hadamard

J. Statist.

rows and columns Vol. 1036. Springer, matrices

Plann.

Inference

form a group.

4,

Com-

Berlin-Heidelberg-

CiH(4q, EA(q)).

J. Statist. Plann.