View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2001) 22, 1033–1045 doi:10.1006/eujc.2001.0539 Available online at http://www.idealibrary.com on Conference Matrices and Unimodular Lattices ROBIN CHAPMAN Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by nonprincipal ideals yields simple constructions of further lattices including the Leech lattice. c 2001 Academic Press 1. INTRODUCTION We use conference matrices to define an action of the complex numbers on the real Euclidean n vector space R . In certain cases, the lattice DnC becomes a module over a ring of quadratic in- tegers. We can then obtain new unimodular lattices, essentially by multiplying the lattice DnC by a nonprincipal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the Leech lattice, can be constructed in this way. Recall that a lattice 3 is a discrete subgroup of a finite dimensional real vector space V . We suppose that V has a given Euclidean inner product .u; v/ u v and the rank of 3 equals the dimension of V . In this case 3 has a bounded fundamental7! · region in V . We call the volume of such a fundamental region (measured with respect to the Euclidean structure on V ) the volume of the lattice 3. The lattice 3 is integral if u v Z for all u, v 3. It is even if u 2 u u 2Z for all u 3. Even lattices are necessarily· 2 integral. The lattice2 3 is unimodularj j Dif 3· is2 integral and has2 volume 1. It is well known [9, Chapter VIII, Theorem 8] that if 3 is an even unimodular then the rank of 3 is divisible by 8. For convenience we call the square of the length of a vector its norm. The minimum norm of a lattice is the smallest nonzero norm of its vectors. 2. CONFERENCE MATRICES Let l be a positive integer. A conference matrix of order n [7, Chapter 18] is an n-by-n matrix W satisfying (a) the diagonal entries of W vanish, while its off-diagonal entries lie in 1; 1 , (b) WW .n 1/I , where I denotes the n-by-n identity matrix. {− g > D − Let n denote the set of skew-symmetric conference matrices of order n. W t 2 Let W n. Then H I W satisfies HH .I W/.I W/ I W 2 W D C D C − D − D I WW > nI . As all the entries of H lie in 1; 1 then H is a Hadamard matrix. ConsequentlyC D [7, Theorem 18.1] n 1, 2 or is a multiple{− ofg 4. D n Suppose that n is a multiple of 4 and let l n 1. Fix W n and let V R denote the n-dimensional real vector space under the standardD − Euclidean2 W dot product. WeD shall make V into a vector space over the complex numbers. Each complex number z has the form r sp l where r and s are real numbers and p l ipl. We then define C − − D r sp l v v.r I sW/ rv svW C − D C D C for each v V . Then, since W 2 l I , it is straightforward to show that V is now a complex vector space,2 that is D − 0195–6698/01/081033 + 13 $35.00/0 c 2001 Academic Press 1034 R. Chapman (a) z.v w/ zv zw for z C, v, w V ; C D C 2 2 (b) .z1 z2/v z1v z2v and .z1z2/v z1.z2v/ for z1, z2 C, v V . C D C D 2 2 As a complex vector space, V has dimension n=2. Let v pv v denote the Euclidean length of a vector v Rn. This action of C on n j j D · 2 R transforms lengths in a simple way. Let z∗ denote the complex conjugate of the complex number z. n LEMMA 2.1. (a) If z1, z2 C and v1, v2 R then .z1v1/ .z2v2/ .z1z2∗v1/ v2 (b) If z C and v Rn then2zv z v . 2 · D · 2 2 j j D j jj j PROOF. Let z j r j s j p l with r j , s j R. Then D C − 2 .z1v1/ .z2v2/ .z1v1/.z2v2/> · D v1.r1 I s1W/.r2 I s2W/>v> D C C 2 v.r1 I s1W/.r2 I s2W/v> D C − v..r1r2 ls1s2/I .s1r2 r1s2/W/v> D C C − .z1z∗v1/ v2 D 2 · as claimed. Consequently 2 2 2 2 zv .zv/ .zv/ .zz∗v/ v z v v z v : 2 j j D · D · D j j · D j j j j Thus for fixed nonzero z, the map v zv is a similarity of Rn with scale factor z . 7! j j 3. QUADRATIC FIELDS We retain the previous notation. Suppose in addition that l n 1 is square free. Let K denote the quadratic field Q p l . Since l is square free, the ringD of− integers of K is − 1 p l a bp l Z " C − # ( C − a; b Z; a b .mod 2/) : O D 2 D 2 V 2 ≡ In particular is a Dedekind domain. We shall show that some of the familiar lattices L in l 1 O R C are modules for the ring , that is that αL L whenever α . Let O ⊆ 2 O n L0 .a1;:::; an/ Z a1 an 0 .mod 2/ D f 2 V C···C ≡ g be the Dn root lattice. LEMMA 3.1. The lattice L0 is an -module. O 1 1 PROOF. It suffices to show that 1 p l v v.I W/ L0 whenever v L0. Indeed 2 C − D 2 C 2 2 it suffices to show this whenever v lies in a generating set for L0. Now L0 is generated by the vectors 2e j and e j ek (for j k) where e j denotes the jth unit vector. Firstly e j .I W/ is a row of the HadamardC matrix6DI W. As it contains n instances of 1 and n is even,C it lies 1 C ± in L0. Next 2 .e j ek/.I W/ is the sum of two rows of the Hadamard matrix I W. Two C C 1 C rows of an n-by-n Hadamard matrix agree in exactly n=2 places. Hence 2 .e j ek/.I W/ 1 C C has n=2 zeros and n=2 instances of 1. As n=2 is even then 2 .e j ek/.I W/ L0. This completes the proof. ± C C 2 2 Unimodular lattices 1035 Now consider the set .a1;:::; an/ a j 1=2; 1=2 : S D f V 2 {− gg The difference of two vectors in lies in L0 if and only if those vectors agree in an even S number of places. Thus there are exactly two cosets v L0 as v runs through . The union of with either of these cosets is a lattice and it will beC convenient to distinguishS these lattices byS a notation dependent on the matrix W. 1 1 For each j, 2 e j .I W/ , and for each j and k, 2 .e j ek/.I W/ L0 by Lemma 3.1. 1 C 2 S − C 2 Thus the cosets e j .I W/ L0 are identical. Let 2 C C 1 v v e1.I W/ L0 SC D f 2 S V − 2 C 2 g and : S− D S n SC 1 1 1 As e j .I W/ e j . I W/ e j = L0 then e j . I W/ for each j. 2 C − 2 − C D 2 2 − C 2 S− If v then 2v has n entries 1 and so 2v L0. It follows that L0 .v L0/ is a lattice, 2 S ± 2 [ C which depends only on whether v or v . We write L for L0 .v L0/ when 2 SC 2 S− C [ C v and L for L0 .v L0/ when v . Both L and L are isometric to the lattice 2 SC − [ C 2 S− C − usually denoted by DnC [5, Chapter 4, Section 7.3]. The lattice DnC is unimodular for each n divisible by 4, and it is even unimodular whenever n is divisible by 8. LEMMA 3.2. If n is divisible by 8 then the lattices L and L are -modules. C − O PROOF. Let L L or L . Then L L0 .v L0/ for some v and by Lemma 3.1 D C1 − D1 C C 1 2 S it suffices to show that 2 1 p l v 2 v.I W/ L. Note that 4 .l 1/ is an even integer by the hypothesis. C − D C 2 C 1 1 We may assume that v e1. I W/. If v e1.I W/ then D 2 ± C D 2 C 1 1 2 1 1 l 1 v.I W/ e1.I W/ e1..1 l/I 2W/ e1.I W/ C e1 2 C D 4 C D 4 − C D 2 C − 4 1 which lies in L as 2 e1.I W/ L. 1 C 2 If v e1.