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rundlehrender
mathematischen Wissenschaften 29
Series o Comprehensive Studies in Mathematics
Editors
S.S. Chern B. Eckmann
P
de la Harpe
H Hironaka
F
Hirzebruch N Hitchin
L Hormander M. A. Knus A Kupiainen
J
Lannes G Lebeau M Ratner D Serre
Ya.G. Sinai N
J
A Sloane
J
Tits
M
Waldschmidt S. Watanabe
Managing Editors
M
Berger J Coates S.R.S. Varadhan
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Springer Science Business Media, LLC
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J.H. Conway N .J.A. Sloane
Sphere Packings
Lattices and Groups
Third Edition
With Additional Contributions by
E. Bannai R.E. Borcherds
J
Leech
S.P. Norton A.M. Odlyzko R.A. Parker
L. Queen and B. B. Venkov
With
112
Illustrations
Springer
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J.H.
Conway
Mathematics Department
Princeton University
Princeton, NJ 08540
USA
conway@
math.
princeton.edu
N.J.A.
Sloane
Information Sciences Research
AT T Labs - Research
180 Park Avenue
Florham
Park,
NJ
07932 USA
Mathematics Subject
Classification (1991): 0 5B40, 11H06, 20E32, 11T71, 11E12
Library
of
Congress Cataloging-in-Publication Data
Conway, John Horton
Sphere packings, lattices and
groups.-
3rd
ed./
J.H. Conway,
N.J.A. Sloane.
p em. - Grundlehren der mathematischen Wissenschaften ;
290)
Includes bibliographical references and index.
ISBN 978-1-4419-3134-4 ISBN 978-1-4757-6568-7 eBook)
DOI 10.1007/978-1-4757-6568-7
1 Combinatorial packing and covering. 2 Sphere. 3 Lattice
theory. 4 Finite groups.
I
Sloane, N.J.A. Neil James
Alexander).
1939-
. II Title. III Series.
QA166.7.C66 1998
511'.6-dc21
98-26950
Printed on acid-free paper.
springeronline.com
1999, 1998, 1993 Springer Science+Bsiness Media New York
Originally published by Springer-Verlag New York, Inc. in 1999
Softcover reprint of he
hardcover 3rd
edition 1999
All rights reserved. This work may not be translated or copied in whole
or
in part without the
written permission
of
the publisher Springer Science+Business Media, LLC,
except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or here
after developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if
the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
by anyone.
Production managed by Allan Abrams; manufacturing supervised by Thomas King.
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P
reface t
o First
dition
The main t
hem es.
This book is
mainly conc
erned with th
e problem o
f
packing sp
heres in
Euclidean sp
ace of dimens
ions
1 2
3 4
5,.... Given
a
la rge n
umber ofequ
al spheres, w
hat
is
th e mos
t efficient (or
densest) way
to pack th
em together?
W e also stu
dy several clo
sely related p
roblems: the
ki
ssing number
problem
which asks how
many sp h
eres can be a
rranged
so
that they
all touch one
central sp h
ere of the s
ame size; th e
OYering
problem
which ask
s for the leas
t dense way
to cover n-dim
ensional sp a
ce
w
ith equal ov
erlapping sph
eres; and th e
qua
ntizing proble
m im por
tant
for
a
pplications to
analog-to-di
gital conversi
on (or d ata
compression),
which
a
sks
how
to pl
ace points in
space
so
tha
t the average
second mom
ent of
their Vor
onoi cells
is
as small as
possible. A t
tacks on thes
e problems
usually a
rrange the sp
heres so thei
r centers form
a lattice.
Lattices are
desc
ribed by quad
ratic form s
and we study
the classification
o quadratic
form
s. M ost
of the book
is
devoted to t
hese
five
prob
lems.
Th
e miraculous
enters: the
E
8
nd
Leech
la ttices.
When we
investigate
those prob
lems, some
fantastic th i
ngs happen
There are
two sphere
packings, one
in
eig ht dimensions, the
8
lattice and one
in
twenty-four
dimen s
ions, the Leec
h lattice A
2
4
which are u
nexpectedly g
ood and very
sy
mmetrical pa
ckings, and h
ave a num be
r of remarka
ble and myst
erious
properties
, not all of w
hich are com
pletely unde
rstood even t
oday. In a
certa
in sense we
could say th a
t the book
is
devoted to s
tudying these
two
Ia tt
ices and their
properties.
A t on
e point while
working on th
is book we e
ven considered
adopting a
special ab
breviation fo
r
It is
a rem
arkable fact
that", since
this phrase
seem
ed to occur
so often. Bu
t
in
fact we h
ave tried to a
void such ph
rases
and to
maintain a
scholarly dec
orum of langu
age.
Nevertheless
there are a
number of ast
onishing resu
lts in the book
, and
per
haps this is a
good place to
mention som
e of the most
miraculous.
(The
technical te
rms used here
are all define
d later in th e
book.)
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viii
Preface to First Edition
with determinant 2 up to dimension 18 and determinant 3 up to dimension
17 (Chap. 15).
- A simple description of a construction
for
the Monster simple group
(Chap. 29).
Other tables which up to now could only be found in journal articles or
conference proceedings, include:
- Bounds for kissing numbers in dimensions up to 24 (Table
1.5).
- The Minkowski-Siegel mass constants for
even
and odd unimodular
lattices in dimensions up to 32 (Chap. 16).
- The
even
and odd unimodular lattices in dimensions up to 24 (Table 2.2
and Chaps. 16, 17).
- Vectors in the first eight shells of the
8
lattice (Table 4.10) and the
first three shells of the Leech lattice (Table 4.13).
- Best's codes of length
10
and
11,
that produce the densest packing
known P
10
c
in 10 dimensions and the highest kissing number known
c in 11 dimensions (Chap.
S).
- Improved tables giving the best known codes of length 2m for
m
8
(Table 5.4) and of all lengths up to 24 (Table 9.1).
- Laminated lattices in dimensions up to 48 (Tables 6.1, 6.3).
- The best integral lattices of minimal norms 2, 3 and 4, in dimensions up
to
24
(Table 6.4).
- A description of E
8
lattice vectors in terms of icosians (Table 8.1).
- Minimal vectors in McKay's 40-dimensionallattice M
40
(Table 8.6).
- The classification of subsets of
24
objects under the action of the
Mathieu group M
24
(Fig. 10.1).
- Groups associated with the Leech lattice (Table 10.4).
- Simple groups that arise from centralizers in the Monster (Chap. 10).
- Second moments of polyhedra in 3 and 4 dimensions (Chap. 21).
The deep holes in the Leech lattice (Table 23.1).
- An extensive table of Leech roots, in both hyperbolic and Euclidean
coordinates (Chap. 28).
- Coxeter-Vinberg diagrams for the automorphism groups of the lattices
In, for n
20
(Chap. 28) and IIn,l for n 24 (Chap. 27).
The contents
o
the chapters.
Chapters
1-3
form an extended introduction
to the whole book. In these chapters
we
survey what is presently known
about the packing, kissing number, covering and quantizing problems.
There are sections on quadratic forms and their classification, the
connections with number theory, the channel coding problem, spherical
codes, error-correcting codes, Steiner systems, t-designs, and the
connections with group theory. These chapters also introduce definitions
and terminology that will be used throughout the book.
Chapter 4 describes a number of important lattices, including the cubic
lattice
zn
the root lattices An, Dn. E
6
, E
7
, E
8
,
the Coxeter-Todd lattice
K
,
the Barnes-Wall lattice A
16,
the Leech lattice A
24,
and their duals.
Among other things we give their minimal vectors, densities, covering radii,
glue vectors, automorphism groups, expressions for their theta series, and
tables of the numbers of points in the first fifty shells. We also include a
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Preface
to
First Edition ix
brief discussion of reflection groups and of the technique of gluing lattices
together.
Chapters 5-8 are devoted to techniques for constructing sphere
packings. Many of the constructions in Chaps. 5 and 7 are based on
error-correcting codes; other constructions in Chapter 5 build up packings
by layers. Layered packings are studied in greater detail in Chap. 6,
where the formal concept of a
laminated lattice
n is introduced. Chapter
8 uses a number of more sophisticated algebraic techniques to construct
lattices.
Chapter 9 introduces analytical methods for finding bounds on the best
codes, sphere packings and related problems. The methods use techniques
from harmonic analysis and linear programming. We give a simplified
account of Kabatiansky and Levenshtein's recent sphere packing bounds.
Chapters 10 and
11
study the Golay codes of length 12 and 24, the
associated Steiner systems S 5,6,12) and S 5,8,24), and their
automorphism groups M
12
and M
24
. The
MINIMOG
and MOG (or
Miracle Octad Generator) and the Tetracode and Hexacode are
computational tools that make it easy to perform calculations with these
objects. These two chapters also study a number of related groups, in
particular the automorphism group (or O of the Leech lattice. The
Appendix to Chapter 10 describes all the sporadic simple groups.
Chapter 12 gives a short proof that the Leech lattice is the unique even
unimodular lattice with
no
vectors of norm
2.
Chapter
13
solves the
kissing number problem in 8 and 24
dimensions - the
E
8
and Leech
lattices have the highest possible kissing numbers in these dimensions.
Chapter
14
shows that these arrangements of spheres are essentially
unique.
Chapters 15-19 deal with the classification of integral quadratic forms.
Chapters
16
and
18
together give three proofs that Niemeier's enumeration
of the 24-dimensional even unimodular lattices is correct. In Chap.
19
we
find all the extremal odd unimodular lattices in any dimension.
Chapters 20 and
21
are concerned with geometric properties of lattices.
In Chap. 20
we
discuss algorithms which, given an arbitrary point
of
the
space, find the closest lattice point. These algorithms can be used for
vector quantizing or for encoding and decoding lattice codes for a
bandlimited channel. Chapter
21
studies the Voronoi cells of lattices and
their second moments.
Soon after discovering his lattice, John Leech conjectured
that
its
covering radius was equal to J2 times its packing radius, but was unable to
find a proof. In 1980 Simon Norton found an ingenious argument which
shows that the covering radius is no more than 1.452... times the packing
radius (Chap. 22), and shortly afterwards Richard Parker and the authors
managed to prove Leech's conjecture (Chap. 23).
Our method of proof involves finding all the deep holes in the Leech
lattice, i.e. all points of 24-dimensional space that are maximally distant
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X
Preface to First Edition
from the lattice. We were astonished to discover that there are precisely
23 distinct types of deep hole, and that they are in one-to-one
correspondence with the Niemeier lattices the 24-dimensional even
unimodular lattices of minimal norm 2) - see Theorem 2 of Chap. 23.
Chapter 23, or the Deep Holes paper, as it is usually called, has turned out
to be extremely fruitful, having stimulated the remaining chapters in the
book, also Chap. 6, and several journal articles.
In Chap.
24 we
give
23
constructions for the Leech lattice, one for each
of the deep holes or Niemeier lattices. Two of these are the familiar
constructions based on the Golay codes. In the second half of Chap. 24
we
introduce the
hole diagram
of a deep hole, which describes the environs of
the hole. Chapter 25 the Shallow Holes paper) uses the results of
Chap.
23
and 24 to classify
all
the holes in the Leech lattice.
Considerable light is thrown on these mysteries by the realization that
the Leech lattice and the Niemeier lattices can all be obtained very easily
from a single lattice, namely
11
25
,
the unique even unimodular lattice
in
Lorentzian space R
25
1
For any vector w E R
25
1
,
let
w L
=
{x
Ellz5,
:xw=O}.
Then if w
is
the special vector
w
25
=
0,1,2,3, . . .
,23,24170 ,
w L/w
is
the Leech lattice, and other choices for
w
lead to the 23 Niemeier
lattices.
The properties of the Leech lattice are closely related to the geometry
of the lattice
25
,
1
The automorphism groups .of the Lorentzian lattice
In,l for n
~
19 and IIn,l for n = 1 9 and 17 were found by Vinberg,
Kaplinskaja and Meyer. Chapter
27
finds the automorphism group of
2
s,i This remarkable group has a reflection subgroup with a Coxeter
diagram that is, speaking loosely, isomorphic to the Leech lattice. More
precisely, a set of fundamental roots for
25
,
1
consists of the vectors
r E
ll2s,1
satisfying
and we call these the Leech roots Chapter 26 shows that there is an
isometry between the set of Leech roots and the points of the Leech lattice.
Then the Coxeter part of the automorphism group of
25
,
1
is
just the
Coxeter group generated by the Leech roots Theorem I of Chap. 27).
Since
25
,
1
is
a natural quadratic form to study, whose definition
certainly does not mention the Leech lattice, it
is
surprising that the Leech
lattice essentially determines the automorphism group of the form.
The Leech roots also provide a better understanding of the
automorphism groups of the other lattices
In l
and
lln,h
as
we
see
in
Chap. 28. This chapter also contains an extensive table of Leech roots.
Chapter 29 describes a construction for the Monster simple group, and the
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xii Preface to First Edition
published by Academic Press, N.Y. [Con13]; Chap. 28 on J.H.C. and
N.J.A.S.,
Proc
Royal Society London A384 (1982) [Con33]; Chap. 30 on
R. E. Borcherds, J.H.C., L Queen and N.J.A.S.,
Advances in Math.
53
(1984), published
by
Academic Press, N.Y. [Bor51.
Our collaborators mentioned above are:
Eiichi Bannai, Math. Dept., Ohio State University, Columbus, Ohio 43210;
Richard E. Borcherds, Dept. of Pure Math. and Math. Statistics,
Cambridge University, Cambridge CB2 lSB, England;
John Leech, Computing Science Dept., University of Stirling, Stirling FK9
4LA, Scotland;
Simon P.
Norton, Dept. of Pure Math. and Math. Statistics, Cambridge
University, Cambridge CB2 lSB, England;
Andrew M. Odlyzko, Math. Sciences Research Center, AT T Bell
Laboratories, Murray Hill, New Jersey 07974;
Richard
A.
Parker, Dept. of Pure Math. and Math. Statistics, Cambridge
University, Cambridge CB2 lSB, England;
Larissa Queen, Dept. of Pure Math. and Math. Statistics, Cambridge
University, Cambridge CB2 lSB, England;
B. B. Venkov, Leningrad Division of the Math. Institute of the USSR
Academy of Sciences, Leningrad, USSR.
Acknowledgements. We thank all our collaborators and the publishers of
these articles for allowing us to make use of this material.
We should like to express our thanks to
E.
S. Barnes,
H.
S. M. Coxeter,
Susanna Cuyler,
G. D.
Forney, Jr.,
W.
M.
Kantor,
J. J.
Seidel, J.-P. Serre,
P. N. de Souza, and above all John Leech, for their comments on the
manuscript. Further acknowledgements appear at the end o the individual
chapters. Any errors that remain are our own
responsibility: please
notify N. J. A. Sloane, Information Sciences Research, AT T Labs -
Research, 180 Park Avenue, Florham Park, NJ 07932-0971, USA (email:
[email protected]). We would also like to hear o any improvements
to the tables.
We thank Ann Marie McGowan, Gisele Wallace and Cynthia Martin,
who typed the original versions of many of the chapters, and especially
Mary Flannelly and Susan Tarczynski, who produced the final manuscript.
B.
L English and R. A. Matula of the Bell Laboratories library staff
helped locate obscure references.
N.J.A.S. thanks Bell Laboratories (and especially R. L Graham and
A. M. Odlyzko) for support and encouragement during this work, and
J.H.C. thanks Bell Laboratories for support and hospitality during various
visits to Murray Hill.
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Preface t First Edition
xiii
We remark that in two dimensions the familiar hexagonal lattice
0
0
0 0
solv s
the packing kissing covering and quantizing problems. In a sense
this whole book is simply a search for similar nice patterns
in
higher
dimensions.
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Prefac
e to Thir
d dition
Interest in
the subject
matter o th
e book con
tinues to gro
w. The
Sup
plementary B
ibliograph y ha
s been enlarg
ed
to
cover th
e period 198
8 to
1998
a
nd now cont
ains over 800
items. Oth
er changes fr
om the secon
d
edition incl
ude a handfu
l
o
sm all co
rrections and
improvemen
ts to the
main te
xt, and this
preface (an
expanded ver
sion
o
the p
reface to th
e
Second
Edition) whic
h contains a b
rief report on
some o the
developments
sin
ce the appear
ance o the
first edition.
W
e are grat
eful to a nu
mber
o
corr
espondents w
ho have su p
plied
corre
ctions and co
mments on th
e first two e
ditions, or w
ho have sent
us
copies o m
anuscripts.
We thank in p
articular R. B
acher, R. E.
Borcherds,
P.
Boy
valenkov, H.
S. M. Coxet
er,
Y.
Edel, N. D.
Elkies,
L.
J.
Gerstei
n,
M
.
H
arada, J. Lee
ch,
J.
H. Lin
dsey, II, J. M
artinet, J. M
cKay, G. Neb
e,
E.
Pervin, E. M.
Rains, R.
Scharlau, F.
Sigrist, H.
M. Switkay, T.
Urabe,
A. Vardy
Z.-X. Wan
and J Wills
. The new m
aterial was e
xpertly typed
by
Susan K. Po
pe.
We are
planning a
sequel, tent
atively entitle
d
The
Geometry
o
Low-Dim
ensional
roups
and Lattices which will include two earlier
papers
[Con36] and
[Con37] no
t included i
n this book,
as well as
several r
ecent papers
dealing with
groups and
lattices in low
dimensions
( [CSL D L
IHC SL D L8]
, [CoS19la],
[CoS195a], et
c.).
A
Russian versi
on
o
the first
edition, trans
lated by S. N.
Litsyn, M. A.
Tsfasman an
d
G. B.
Sha
bat, was publ
ished by Mir
(Moscow) in
1990.
Recent de
velopments c
omments and
additional c
orrections.
The fo
llowing
pa
ges attempt to
describe rec
ent developm e
nts
in
some o
the topics t
reated
in t
he book. Th
e arrangemen
t roughly fol
lows that o
the chapters.
Our
We als
o thank the c
orrespondent
who reported
hearing the f
irst edition
described
during a tal
k
as
th e bib
le
o
the sub
ject, and, lik
e the bible,
[it] con
tains no proo
fs . This is o
course on l
y half true.
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Prefuce
to Third Edition
xvii
fill in the details, but I feel that the greater part of the work has yet
to be done. Hsiang [Hsi93b] also claims to have a proof that no more
than 24 spheres can touch an equal sphere
in
four dimensions. For further
discussion see [CoHMS], [Hal94], [Hsi95].
K. Bezdek [Bez97] has made some partial progress towards solving
the dodecahedral conjecture This conjecture, weaker than the Kepler
conjecture, states that the volume of any Voronoi cell in a packing of unit
spheres in
3
is at least as large as the volume of a regular dodecahedron
of inradius
1.
See also Muder [Mude93].
A.
Bezdek, W. Kuperberg and Makai [BezKM91] had established the
Kepler conjecture for packings composed
of
parallel strings
of
spheres.
See also Knill
[Kni1196].
There was no reason to doubt the truth
of
the Kepler conjecture.
However,
A.
Bezdek and W. Kuperberg [BezKu91] show that there are
packings of congruent ellipsoids with density 0.7533 exceeding rr/.JTii,
and in [Wills91] this is improved to 0.7585
Using spheres
of
two radii 0
s) ~
2\n Ins In In Ins(I o(l)).
Plesken [Plesk94] studies similar embedding questions for lattices from
a totally different point of
view.
See also Cremona and Landau [CrL90].
Complexity
For recent results concerning the complexity
of
various lattice- and coding
theoretic calculations (cf.
1.4
of Chap. 2), see
Ajtai [Ajt96], [Ajt97],
Downey et
al.
[DowFV], Hastad [Has88], Jastad and Lagarias [HaL90],
Lagarias [Laga96], Lagarias, Lenstra and Schnorr [Lag3],
Paz
and Schnorr
[PaS87],
Vardy [Vard97].
In
particular,
Vardy [Vard97]
shows that computing the minimal
distance of a binary linear code
is
NP-hard,
and
the corresponding decision
problem
is
NP-complete. Ajtai [Ajt97] has made some progress towards
establishing analogous results for lattices. Downey
et
al. [DowFV] show
that computing (the nonzero terms in) the theta-series
of
a lattice
is
NP-hard.
For
lattice reduction algorithms see also
[Schn87], [Val90], [Zas3].
Most
of
these results assume the lattice in question
is
a sublattice
of
zn.
In
this regard the results of [CSLDL5] mentioned above are especially
relevant. Ivanyos and Szanto [IvSz96] give a version
of
the LLL algorithm
that applies to indefinite quadratic forms.
Mayer [Maye93], [Maye95] shows that every Minkowski-reduced basis
for a lattice of dimension n ~ 6 consists of strict
Voronoi
vectors (cf.
[Rys8]).
He
also answers a question raised by Cassels ([Cas3],
p.
279)
by showing that
in
seven dimensions (for the first time) the Minkowski
domains do not meet face to face.
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XXX
Prefac
e to Third Edit
ion
its theta
series. Thus
n-dimensional
isospectral
lattices exist
if and only
if n is at
least
4
. For
more about t
hese matters
see [CoFu97]
.
a
ttice quantize
rs
Cou
lson [Coul91]
has found th
e mean s ~ u a r e d
error G fo
r the perfect
and
isodual)
six-dimensiona
l lattice ~
A
6
2
l define
d
in
6
of C
hap. 8 and
studied in [
CSLDL3]). H
e finds G =
0.075057, giv
ing
a
n additi
onal entry
for
Table 2.3 of Ch
ap.
2.
Viterb
o and Biglieri
[ViBi96] hav
e computed
G
for the
lattices of
Eqs. 1) and
2), the Dicks
on lattices o
f page 36, and
other la
ttices.
Ag
rell and Erik
sson [AgEr9
8] have foun
d 9- and
10-dimensiona
l
lattices with G
=
0.0716
and
0.0708, respectively,
and
show that the
nonlattic
e packings
D ~ and t
hav
e G =0.0727
and 0.0711,
respectively.
Th
ese values ar
e all lower i
.e. better) tha
n the previou
s records.
Notes
on Chapte
r : Codes
Designs a
nd Groups
Latti
ce codes
Several au
thors hav
e
stu
died the erro
r probability
of codes
for the
Gaussian cha
nnel that ma
ke use of co
nstellations o
f points from
some lattice
as
the signal set - see for example Banihashemi
and
Khandani
[BanKh9
6], de Buda
[Bud2], [Bu
d89], Forney
[Forn97], L
inder et al.
[LiSZ93],
Loeliger [Loe19
7], Poltyre
v [Polt94], T
arokh,
Vardy and Zeger
[TaVZ],
Urbanke and
Rimoldi [UrB
98].
Urban
ke and Rimo
ldi [UrB98],
completing th
e work of se
veral others,
ha
ve shown that
lattice codes
bounded by a
sphere can a
chieve the ca
pacity
V
log
2
I
PIN where P
is
the sign
al power a
nd N is the
noise
variance),
using minima
l-distance dec
oding. This is
stronger tha
n what can
b
e deduce
d directly
from the Minko
wski-Hiawka
theorem [Bud
2], [Cas2],
[Grul
a], [H lal], [
Rog7]), whic
h
is
that a
rate of
1
h_log
2
PIN
can
be
achie
ved with latt
ice codes.
There
has been a g
reat deal of
activity
on
trellis
codes
cf.
1.4
of Chap. 3
- see for
example [BD
MS], [Ca191
], [Ca090],
[Forn88],
[Forn88a], [F
orn89a], [For
n91], [FoCa89
], [FoWe89],
[LaVa95], [L
aVa95a],
[LaVa96], [T
aVa97], [VaKs9
6].
Another
very interestin
g question is
that of findin
g
tre
llis represent
a-
tions
of the s
tandard codes
and lattices:
see Forney
[Forn94], [Fo
rn94a],
F
eigenbaum et
al. [FeFMMV
], Var
dy
[V
ard98a] and m
any related p
apers:
[BanB196], [Ban
Kh97], [B1
Ta96], [FoTr93],
[KhEs97], [T
a8196], [TaB196
a].
W
e
have already
mentioned
recent work on
Goppa c
odes and the
c
onstruction of
codes and l
attices fr
om
a
lgebraic geom
etry cf. 2
.11 of
Ch
ap. 3 under 1.5
of Chap
. l
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xxxii
Preface
to
Third Edition
geometric description of the maximal irreducible subgroups of GL n, Z)
for
n
= 1, . . . , 9, II
13,
17, 19, 23, by exhibiting lattices corresponding to
these quadratic forms (cf. 4.2(i) of Chap. 3): the automorphism groups
of the lattices are the desired groups. By giving natural coordinates for
these lattices and determining their minimal vectors, we are able to make
their symmetry groups clearly visible. There are 176 lattices, many
of
which have not been studied before (although they are implicit
in
the
above references and
in
[Conl6]).
The book by Holt and Plesken [HoP189] contains tables of perfect
groups of
order up to 10
6
,
and
includes tables
of
crystallographic space
groups
in
dimensions
up
to
10.
Nebe and Plesken [NeP195] and Nebe [Nebe96], [Nebe96a] (see also
[Plesk96], [Nebe98a]) have recently completed the enumeration
of
the
maximal finite irreducible subgroups of G L n, Q) for n ::: 31, together
with the associated lattices. This is
an
impressive series of papers, which
contains
an
enormous amount of information about lattices in dimensions
below 32.
Notes
on
hapter : ertain Important Lattices and Their
Properties
Several recent papers have dealt with gluing theory (cf.
3
of
Chap.
4
and related techniques for combining lattices: [GaL91]-{GaL92a],
[Gers91], [Sig90], [Xul]. Gannon and Lam [GaL92], [GaL92a] also give
a number of new theta-function identities (cf.
4.1
of Chap. 4).
Scharlau and Blaschke [SchaB96] classify all lattices
in
dimensions
n ::: 6
in
which the root system has full rank.
Professor Coxeter has pointed out to us that, in the last line of the
text on page 96, we should have mentioned the work of Bagnera
[BagOS]
along with that of Miller.
For recent work on quatemionic reflection groups (cf. 2 of Chap. 4)
see Cohen [Coh91].
Hexagonal lattice A
The number of inequivalent sublattices of index N in A
2
is determined
in [BerS197a], and the problems of determining the best sublattices from
the points
of
view
of
packing density, signal-to-noise ratio and energy are
considered. These questions arise in cellular radio. See also [BaaP195].
Kiihnlein [Kuhn96] has made some progress towards establishing
Schmutz s conjecture [Schmu95] that the distinct norms that occur in
A
2
are strictly smaller than those in any other (appropriately scaled)
two-dimensional lattice. See also Schmutz [Schmu93l, Schmutz Schaller
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xxxiv
Preface to Third Edition
Hall-Janko group
1
2
(cf. Chap. 10 . The density
of
this lattice is about
a quarter
of
that
of
the Leech lattice.
Napias [Napa94] has found
some new
lattices by investigating cross
sections of the Leech lattice, the 32-dimensional Quebbemann lattice and
other lattices.
Shadows and parity or characteristic) vectors
The notion of the shadow of a self-dual code or unimodular lattice,
introduced
in [CoS190],
[CoS190a], has proved useful
in
several contexts,
and if we were to rewrite Chapter 4 we would include the following
discussion there.
We
will concentrate
on
lattices, the treatment for codes
being analogous.
Let A
be an
n-dimensional odd unimodular (or Type I lattice, and let
A
0
be the even sublattice, of index 2. The dual lattice ~ is the union
of
four cosets
of
A
0
, say
~ = A
0
U A, U A
2
U
A3
where A = A
0
U A
2
Then we call S := A
1
U A
3
= A ~ \
A the shadow
of A.
If
A is even (or Type
II
we define its shadow S to be A itself.
The following properties are easily established [CoS190].
If
s E
S and
X E
A, then
s
X
E
Z if
X E
A
0
, s X E tz \
Z if
X E
A
1
In fact the set 2S = {2s : s E
S} is
precisely the set of parity vectors for
A, that
is,
those vectors
u
E
A such that
U
X = X X (mod 2
for
all
X E
A .
Such vectors have been studied by many authors, going back at least as far
as
Braun [Brau40] (we thank H.-G. Quebbemann
for
this remark). They
have been called characteristic vectors [Blij59], [Borl], [Elki95], [Elki95a],
[Mil7], canonical elements [Serl],
and
test vectors.
We
recommend parity
vector
as
the standard name for this concept.
The existence
of
a parity vector
u
also follows from the fact that the
map
x
~
x
x (mod 2)
is
a linear functional from A
to
lF
2
. The set 2S
of
all parity vectors forms a single class
u
2A
in
A/2A.
If
A
is
even
this
is
the zero class.
We also note that for any parity vector
u, u
u
= (mod 8 .
Gerstein [Gers96] gives
an
explicit construction for a parity vector. Let
v
1
, n
be
a basis
for
A and
v;, . . . ,
the dual basis. Then L
c;
v; is
a parity vector if and only if
c; = v
(mod 2
for
all
i.
Elkies [Elki95], [Elki95a] shows that the minimal norm p A) of any
parity vector
for
A satisfies
p A) ~
n
and
p A)
=
n if
and only
if
A =
zn.
Furthermore, if
p A)
= 8 then A = zn r
E9 M ,
where M, is
one of the fourteen unimodular lattices whose components are E
8
, D
12
, E
A,s,
Di
A11E6
~ D
Dl Aj Af
2
,
0
3
(using the notation
of Chapter 16 .
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Preface
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XXX
The shad
ow may also
be defined f
or a more ge
neral class o
f lattices.
If
A is a 2-int
egral lattic
e (i.e. u
v
E :0_ the 2-
adic integers,
for
all
u v
E
A), an
d A
0
= {u E
A : u
u E
2Z
2
}
is the even sub
lattice,
we de
fine the shad
ow S(A) of
A
a
s follow
s [RaS19
8a].
If
A is odd
,
S A)= A
)* \A*, o
therwise
S A)=A*. Then
S(A
) = {v E A IQJ:
2u v
=
u
u (mod
2Z
2
)
for
all
u E
A} .
This i
ncludes the f
irst definition of
shadow as
a special ca
se. The thet
a
series of
the shadow (
for both defin
itions) is rela
ted to the th
eta series of
the lattice
by
(
rr i/4)dimA
( I
E>s A
) Z) = (det A
)
2
ev
z eA
I -
.
(3)
t is also
shown in [
RaS198a] tha
t if A has o
dd determina
nt, then for
u E S(A)
,
u u
= oddity A (m
od 2Z
2
)
4
(
4)
(compare C
hap. 15). In
particular,
if A
is
an odd
unimodular
lattice with
theta
series
[n/8
]
E>A Z
)
= I>r ( :)3(zr -
8
~ s ( z ) '
(5)
r;;
O
(as
in
Eq. (36)
of Chap.
7),
then the theta series of the shadow
is
given
by
(6)
For further i
nformation ab
out the shad
ow theory of
codes and la
ttices
see [
CoS190], [CoS190a],
[CoS198],
[Rain98], [RaS
198], [RaS
I98a]. See
also [Dou9
5]-[DoHa97].
Coordination sequences
Crystallograp
hers speak o
f coordinati
on number
rather than
kissing
number
. Several re
cent papers h
ave investigat
ed the follow
ing generaliza
tion of
this notion (
[BaaGr97], [B
rLa71], [GrB
S], [MeMo7
9], [O'Ke91],
[O'Ke95
]). Let A be
a (possibly n
onlattice) sph
ere packing,
and form an
infinite gr
aph r whose
nodes are th
e centers of th
e spheres and
which has
an
edge for
every pair
of touching
spheres. The
coordination
sequen
ce
of [ with
respect to a
node E [ is
the sequenc
e S O), S(l) ,
S(2),
. .
.
where
S n) is the
number of no
des in [ at
distance n fr
om P (that
is,
such
that the short
est path to P
contains n
edges).
If
A is a la
ttice then the
coordination
sequence is
independent o
f the
cho
ice of P
.
In
[CSLDL7],
extending the
work of O'
Keeffe [O'Ke
91],
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x
xxv
Preface
to
Third
Edition
[O'Ke95],
we detennine
the coordina
tion sequence
s for all the
root lattices
an
d their dual
s. Ehrhart's
reciprocity
law ([Ehr60}
-[Ehr77], [St
an80],
[Stan86])
is used, but t
here are une
xpected comp
lications.
For example,
ther
e are points
in the 11-
dimensional
anabasic latt
ice
of
[CoS1
95],
ment
ioned in the N
otes to Chap
ter I, with th
e property th
at 2Q is clo
ser
to
the orig
in than (in
graph distan
ce).
We
giv
e two exampl
es. For a d-d
imensional la
ttice A it
is convenient
to wr
ite the gener
ating function
S(x) = L:::o
S(n )x
as
)/(
-
x
)
,
where we ca
ll
~ ( x )
the
coordina
tor polynomia
l.
F
or the root l
attice A
it turn
s out that
and for
Es
we
have
P
8
(x)
= I 232x + 722
8x
+ 55384x
3
+ 133510x
4
+ 107
224x
5
+ 2450
8x
6
+ 232x
7
+
.
Thu
s the coord
ination seque
nce
o
b
egins I, 24
0, 9120, 121
680,
864
960, . .
..
For
further examp
les see [BattV
e98], [CSLD
L7], [GrB S],
and
[SloEIS].
We d
o
not k
now the coor
dination seque
nce
of
the L
eech lattice.
I
n
[CS
LDL 7] we a
lso show tha
t among all
the Barlow p
ackings in
three dime
nsions (those
obtained by
stacking
layers, cf. [
CoS195a])
the hexagonal close packing has the greatest coordination sequence, and
the f
ace-centered c
ubic lattice th
e smallest. M
ore precisely
, for any Barl
ow
packi
ng,
On
2 =
S
(n) = [21n
2
2
] 2
(n > 0 .
For an
y n
> I, th
e only Barlo
w packing th
at achieves e
ither the left
hand valu
e or the righ
t-hand value
for all choice
s o central s
phere is the
fa
ce-centered cu
bic lattice o
r hexagonal c
lose-packing,
respectively.
This
interesti
ng result wa
s conjectured
by O'Keeffe
[O'Ke95]; it
had in fact
already bee
n established
(Conway
Sloane 19
93, unpublish
ed notes).
There is
an
assertion
on
p.
801
of
[Hsi93] that
is
equivalent to saying
that a
ny Barlow pa
cking has S(
2) = 44, and
so is plain
ly incorrect:
as
shown in [C
oS195a], ther
e are Barlow
packings wit
h S(2) = 42
, 43 and
44. [CSLDL
7] concludes
with a num
ber
of
open
problems re
la ted to
coordin
ation sequenc
es.
Notes on C
hapter
: Sphere Pa
cking
nd
E
rror Corre
cting
Cod
es
The Ba
rn es W all lat
tices ([Bar18],
6.5 of Chap.
5,
8.1
of C
hap.
8
are
the subject
of
a recent p
aper by Hahn
[Hahn90].
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Preface
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Third Edition
xxxvii
On p. 152 of Chap. 5 we remarked that
it
would be nice to
have
a
list of the best cyclic codes of length 127.
Such
a list has now been
supplied by Schomaker and Wirtz [SchW92]. Unfortunately this does not
improve the
n
=
128
entry
of
Table 8.5. Perhaps someone will now tackle
the cyclic codes of length 255.
The paper by Ozeki mentioned in the postscript to Chap. 5 has now
appeared [Oze87].
Construction
B*
The following construction
is due
to
A. Vardy
[Vard95], [Vard98]
(who gives a somewhat more general formulation). It generalizes the
construction
of
the Leech lattice given
in Eqs.
(135), (136)
of
Chap. 4
and 4.4
of
Chap.
5,
and
we
refer to it
as
Construction
B*
since it can
also
be
regarded as a generalization of Construction B of 3 of
Chap.S
Let 0
=
0
. . .
0 and 1
=
1
. . . 1,
and let
8
and
C
be
n, M, d
binary
codes (in the notation
of
p. 75) such that c
1 +b
= 0 for all b E
8,
c
E
C.
Let A be the sphere packing with centers
0+2b+4x, 1+2c+4y,
where
x
(resp.
y
is any vector
of
integers with an even (resp. odd) sum,
and
b
8, c
E
C. (We regard the components
of
b and c
as
real O s and
I s rather than elements
of
2
. In general A
is
not a lattice.
The most interesting applications arise when d is 7 or 8, in which
case
it
is easily verified that
A has
center density M
7 1
2
j4 (if d
=
7
and
n ~
20) or
M/2 12
(if
d =
8 and
n
~ 24).
Vardy [Vard95], [Vard98] uses this construction to obtain the nonlattice
packings 8;
0
and 8;
7
-
.8;
0
shown in Table I.l. In dimension 20 he uses
a pair
of
(20, 2
9
7)
codes, but
we
will not describe them here since the
same packing will be obtained more simply below. For dimensions 28 and
30 he takes 8 = C L to be the
[28,
14, 8] or [30, IS, 8] double circulant
codes constructed by Karlin (see [Mac6],
p.
509). Both codes contain
1,
are not self-dual, but are equivalent to their duals.
For
n
= 27 we shorten the length 28 code to obtain a [27, 13,
8]
code
A and set 8 =
1 +A,
C = even weight subcode
of A.L.
Similarly for
n =29.
Once
n
exceeds 31,
we
may use Construction D (see Chap. 8, 8)
instead
of
Construction
B*,
obtaining a lattice packing from an [n, k,
8)
code.
In
particular, using codes with parameters [37, 31,
8]
and [38, 22,
8]
(Shearer [Shea88])
we
obtain the lattices
1J
3
1
and
1J
38
mentioned
in
Table
I. I
As
far as
is
known at the present time, codes with parameters
[32,
18,
8], [33,
18,
8], [34,
19,
8],
. . .
, [38, 23, 8], [39, 23,
8]
might exist. If
any one of these could be constructed, a new record for packing density
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Preface
to
Third Edition
xxxix
Notes on Chapter : Laminated Lattices
In 1963
Muses ([Coxl8],
p.
238; [Mus97],
p.
7 discovered that the
highest possible kissing number for a lattice packing
in
dimensions
n
= 0
through 8 (but presumably for no higher n is given by the formula
7)
where
rxl
is the smallest integer ~ X (cf. Table 1.1).
All laminated lattices A in dimensions n ; 25 are known, and their
kissing numbers
are
shown
in Table
6.3.
In
dimensions 26 and above,
as
mentioned on
pp.
178-179, the number of laminated lattices seems
to
be very large,
and
although they all have the same density, we
do
not at
present know the range of kissing numbers that
can
be achieved.
In the mid-1980 s the authors computed the kissing numbers of one
particular sequence of laminated lattices
in
dimensions 26-32, obtaining
values that can be seen in Table 1.2. Because of an arithmetical
error, the value we obtained in dimension 31 was incorrect. Muses
[Mus97] independently studied the (presumed) maximal kissing numbers
of laminated lattices (finding the correct value 202692 in dimension 31)
and has discovered the formulae analogous to (7).
In the Appendix to Chapter
6,
on page
179,
third paragraph, it would
have been clearer
if
we had said that,
for n
;
12,
the integral laminated
lattice A.{3} of minimal norm 3 consists of the projections onto v L of
the vectors of
An l
having even inner product with v, where v
E
An+l
is
a suitable norm 4 vector. For n ; 10,
K.{3}
is defined similarly, using
Kn l instead of
An l
Also A.{3}.L, K.{3}.L denote the lattices orthogonal
to these in A
23
{3}.
A sequel to Plesken and Pobst [Pie6] has appeared - see [Piesk92].
Notes on Chapter 7: Further Connections Between Codes
and Lattices
Upper bounds
The upper bounds on the minimal norm f t of a
unimodular lattice and the minimal distance
d
of a binary self-dual code
stated in Corollary 10 of Chapter 7 have been strengthened. In [RaSI98a]
it is shown that
an
n-dimensional unimodular lattice has minimal norm
(8)
unless n =
23
when f t:::;:
3.
The analogous result
for
binary codes (Rains
[Rain98]) is that minimal distance of a self-dual code satisfies
9)
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Preface
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Third Edition
xli
In the years since the manuscript of [CoS190a] was first circulated,
over 50 sequels have been written, supplying additional examples
of
codes
in the range
of
Table 1.3. In particular, codes with parameters [70, 35,
12]
filling a gap in earlier versions of the table) were found independently by
W. Scharlau and D. Schomaker [ScharS] and M. Harada [Hara97]. Other
self-dual binary codes are constructed in [BrP91], [DoGH97a], [DoHa97],
[Hara96], [Hara97], [KaT90], [PTL92], [Ton89], [ToYo96], [Tsa91], but
these are just a sampling
of
the recent papers see
[RaS198]).
For ternary self-dual and other) codes
see
[Hara98], [HiN88],
[Huff91], [KsP92], [Oze87], [Oze89b], [VAL93].
The
classification of Type I self-dual binary codes of lengths n ~ 30
given in [Plel2] cf.
p.
189
of
Chap.
7)
has been corrected
in
[CoPS92]
see also [Yor89]).
Lam and Pless [LmP90] have settled a question
of
long standing by
showing that there is no [24, 12,
10]
self-dual code over
4
The proof
was by computer search, but required only a few hours of computation
time. Huffman [Huff90] has enumerated some of the extremal self-dual
codes over
4
of lengths 18 to 28.
We
also show in [CoS190], [CoS190a],
[CoS198]
that there are precisely
five
Type
I optimal i.e.
J-L
= J-Ln
lattices in 32 dimensions, but more than
8 x 10
optimal lattices in
33
dimensions; that unimodular lattices with
J-L = 3 exist precisely for
n
: 23,
n
= = 25; that there are precisely three
Type I
extremal self-dual codes
of
length 32; etc.
Nebe [Nebe98] has found an additional example
of an
extremal
unimodular lattice 8n
in
dimension 48, and Bachoc and Nebe
[BacoN98] contruct two extremal unimodular lattices in dimension 80.
One of these
L
80
)
has kissing number 1250172000 see Table 1.2). The
existence
of
an extremal unimodular lattice
in
dimension 72 or
of
an
extremal doubly-even code
of
length 72) remains open.
Several other recent papers have studied extremal unimodular lattices,
especially in dimensions 32, 40, 48, etc. Besides [CoS190], [CoS198],
which
we
have already mentioned,
see
Bonnecaze et
a .
[BonCS95],
[BonS94], [BonSBM], Chapman [Chap 96], Chapman and Sole [ChS96],
Kitazume et
a .
[KiKM], Koch [Koch86], [Koch90], Koch and Nebe
[KoNe93], Koch and Venkov [KoVe89],
[KoVe91],
etc. Other lattices are
constructed in [JuL88].
For doubly-even binary self-dual codes, rasikov and Litsyn [KrLi97]
have recently shown that the minimal distance satisfies
d ~
0.166315 n o n), n oo . 10)
No comparable bound
is
presently known for even unimodular lattices.
For a comprehensive survey of self-dual codes over all alphabets, see
Rains and Sloane [RaS198].
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Preface to Third Edition
xliii
Pobst [FiP96].) The computer languages KANT, PARI and MAGMA (see
the beginning
of
this Introduction) have extensive facilities for performing
such calculations.
The papers [BoVRB, BoV98] give algebraic constructions for lattices
that can be used to design signal sets for transmission over the Rayleigh
fading channel.
Corrections
to Table 8.1
There are four mistakes in Table
8.1.
The entry headed
w
1
K
-
GJH)
should read
1
-2
0
0 0
a
0
0 0
2
0 0 0
0 0
r
-2 0 0
2
and the entry headed wn - (G K H) should read
1
-2 0 0 0
r
-2 0
0
2
a
0 0 0
2
0 0 0 0 0
Further examples o new packings
Dimensions 25 to 30 As mentioned at the beginning
of
Chapter
17,
the 25-dimensional unimodular lattices were classified by Borcherds [Borl].
All
665
lattices (cf. Table 2.2) have minimal norm 1 or 2.
In
dimension 26, Borcherds [Borl] showed that there is a unique
unimodular lattice with minimal norm 3. This lattice, which we will
denote by S
26
, was discovered by J. H. Conway in the 1970's.
The following construction
of
S
26
is a modification
of
one found by
Borcherds. We work inside a Lorentzian lattice P which is the direct sum
of
the unimodular Niemeier lattice and the Lorentzian lattice hi (cf.
Chaps.
16
and 24). Thus P ~ h
6
.I Let p
=
-2, -1. 0, 1, 2) denote the
Weyl vector for A4,
so
that p = p p p p is the
Weyl
vector for
of norm 60, and let v
=
(4, 2 I 9 E hi Then S
26
is the sublattice of P
that is perpendicular to
v
= p
e
v
E
P S
26
can
also be constructed as a
complex 13-dimensional lattice over Q[(l
+
J5)/2] ([Conl6], p. 62).
Here are the properties
of
S
26
It
is a unimodular 26-dimensional
lattice
of
minimal norm 3, center density S =
3I
3
2-
26
= .0237 . . . (not
a record), kissing number 3120 (also not a record), with automorphism
group
of
order 2
8
Y.5
4
.13
=
18720000, isomorphic to Sp
4
(5) (cf. [Conl6],
p. 61; [Nebe96a]). The minimal norm of a parity vector is 10, and there
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xliv Preface to Third Edition
are 624 such vectors. The group acts transitively on these vectors. The
theta series begins
1 +
3120l
+ 102180q
4
+
1482624l
+
. . .
We
do not know the covering radius.
There is a second interesting 26-dimensional lattice, T
16
, an integral
lattice of determinant
3,
minimal norm 4 and center density
lj../3.
This
is
best obtained by forming the sublattice of T
17
see below) that
is
perpendicular to a norm 3 parity vector. T
16
is
of interest because it
shares the record for the densest known packing in 26 dimensions with
the nonintegral) laminated lattices A
26
The kissing number
is
117936
and the group is the same as the group of
T
17
below.
Bacher and
Venkov [BaVe96]
have classified all unimodular lattices in
dimensions
27
and
28
that contain no roots,
i.e.
have minimal norm ~ 3.
In dimension 27 there are three such lattices. In two of them the minimal
norm of a parity vector is 11. These two lattices have theta series
I + 2664l + 101142q
4
+ 1645056q
5
+ . . .
and automorphism groups of orders 7680 and 3317760, respectively. The
third, found
in
[Con16],
we
shall denote by
T
27
It has a parity vector of
norm
3,
theta series
I
1640q
3
+ 119574q
4
+ 1497600q
5
+
and a group
of
order 2
13
3
5
7
2
13
==
1268047872, which is isomorphic to
the twisted group 2
x e
4
2) : 3 [Conl6], p. 89; [Nebe96a]). That this
is the unique lattice with a parity vector of norm 3 was established by
Borcherds [Borl].
The following construction of T
17
is based on the descriptions in
[Conl6], p. 89 and [Borl]. Let V be the vector space of 3 x 3 Hermitian
matrices
y [ ~ ~
] = a,b,c
I
A,B,C),
B A
c
a,
b
c
real ,
over the real Cayley algebra with units ioo
==
1, i
0
, ,
i
6
, in which
i
,
in+
-+
i,
in+
1
-+ j ,
in 4
-+
k generate a quatemion subalgebra for
n
== 0,
. . .
, 6). V has real dimension 3
+
8
x
3
=
27.
We
define an inner
product on V by Norm y)
==
L, Norm yij
.
The lattice T
27
is
generated
by the 3 x 3 identity matrix and the
819
images
of
the norm 3 vectors
[
1 0
0
1
0 0
3)
[
-1 0
OJ
0 0 I
0 I 0
48)
[
0 s s ]
s
- h 11.
s
1
h -
1
h
768)
where
s
=
i
+ i
0
+ +
i
6
)/4, under the group generated by the maps
taking a, b c
I
A, B, C) to a, b c
I
i Ai, iB, iC), b, c a
I
B, C, A) and
a, c b
I
A.,
-C,
-B), respectively, where i e {i
00
, , i
6
}.
These 820
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Prefuce to Third Edition
xlv
norm 3 vectors and their negatives are all the minimal vectors in the
lattice.
The identity matrix and its negative are the only parity vectors
in
T
2
of
norm 3. Taking the sublattice perpendicular
to
either vector gives T
26
,
which therefore has the same group as T
27
In
28 dimensions Bacher and Venkov [Ba
Ve96]
show that there are
precisely 38 unimodular lattices with no roots.
Two of
these have a parity
vector
of
norm 4 and theta series
1
+ 1728l +
106472q
4
+ ...
while for the other 36 the minimal norm
of
a parity vector
is
12 and the
theta series
1
+
2240q
3
+
98280q
4
+ . . .
One of these
36
is the exterior square of
8
,
which has group 2 x G.2,
where G
=
Ot 2) whereas E
8
itself
has
group 2.G.2). One
of
these 36
lattices also appears in Chapman [Chap97].
Bacher [Bace96] has also found lattices 8
27
8
28
, 8
29
in dimensions
27-29 which are denser than the laminated lattices A
27
,
A
29
,
and are
the densest lattices presently known in these dimensions although, as
we
have already mentioned in the Notes to Chapter 5 the densest packings
currently known
in
dimensions 27 to 31 are all nonlattice packings).
Bz
8
can be obtained by taking the even sublattice S
0
of
S
26
,
which has
determinant 4 and minimal norm 4, and finding translates r
0
+ S
0
,
r
1
+S
0
,
r
2
+ S
0
with
r
0
+ r
1
+ r
2
e S
0
and such that the minimal norm
in
each
translate is 3. We then glue S
0
to a copy of
A
2
scaled so that the minimal
norm is
4
obtaining a lattice
8
28
with determinant
3
minimal norm 4,
center density
Ij../3
and kissing number 112458. This is a nonintegral
lattice since the r; are not elements of the dual quotient S
0
S
0
8
29
is
obtained
in
the same way from
T
27
and has determinant 3, minimal norm
4 center density 1/../3 and kissing number 109884.
Dimensions
32 48
56
Nebe [Nebe98] studies lattices in dimension
2 p-l)
on which
SL
2
p) acts
faithfully. For
p =
mod 4) these are cyclotomic lattices over quatemion
algebras. The three most interesting examples given in [Nebe98] are a
32-dimensional lattice with determinant 17
4
, minimal norm
/ L
= 6 center
density 0
= 2-
3
17-
2
=
2.2728
. . .
kissing number
[ =
233376; a 56-
dimensional lattice with det
=
I / L
=
6
o= 3/2f
8
=
85222.69
. . .
r
=
15590400; and a 48-dimensional even unimodular lattice with minimal
norm 6 that is not isomorphic to either P
48
P or P
48
q which we will
denote by P
4
sn
Its automorphism group contains a subgroup
SL
2
13)
whose normalizer
in
the full group
is
an absolutely irreducible group
SL2 B)
S
2
5)).2
2
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Preface t Third Edition
xlvii
This lattice shares with Z,
8
and the Leech lattice the property
of
being
globally irreducible: A/pA is irreducible for every prime
p
However,
Gross [Gro90] remarks that over algebraic number rings
such lattices
are
more common.
He
gives new descriptions
of
several
familiar lattices
as well as
a number
of
new families
of
unimodular
lattices. Further examples
of
globally irreducible lattices have been
found by Gow [Gow89], [Gow89a]. See also Dummigan [Dum97], Tiep
[Tiep91 HTiep97b].
Thompson and Smith actually constructed their lattice by decomposing
the Lie algebra
of
type
8
over C into a family
of
31 mutually
perpendicular Cartan subalgebras. Later authors have used other Lie
algebras to obtain many further examples of lattices, including infinite
families
of
even
unimodular lattices. See Abdukhalikov [Abdu93], Bondal,
Kostrikin and Tiep [BKT87], Kantor [Kant96],
and
especially the book by
Kostrikin and Tiep [KoTi94].
Lattices from tensor products
Much of the final chapter of Kitaoka s book [Kita93] is concerned with
the properties of tensor products of lattices. The minimal norm of a tensor
product L M clearly cannot exceed the product of the minimal norms
of L
and
M,
and may
be
less. Kitaoka says that a lattice
L
is
of E type
if, for any lattice M, the minimal vectors of L M have the form
u
v
for u E
L,
v
EM
(This implies min(LM)
=
min(L) min(M).) Kitaoka
elegantly proves that every lattice
of
dimension
n ; 43 is of
E-type.
On the other hand the Thompson-Smith lattice
T
S
248
is not
of
E-type.
(Thompson s proof: Let L
=
TS
8
, and consider LL :: Hom L, L .
The element
of LL
corresponding
to
the identity element
of Hom L, L)
is
easily seen to have norm 248, which
is
less than the square
of
the
minimal norm of
L.)
Steinberg ([Mil7], p. 47) has shown that there are
lattices in every dimension
n ::: 292
that
are
not
of
E-type.
If an extremal unimodular lattice of dimension
96
(with minimal norm
10) could be found, or an extremal 3-modular lattice in dimension 84 (with
minimal norm
16),
etc., they would provide lower-dimensional examples
of
non-E-type lattices.
Coulangeon [Cogn98] has
given
a generalization
of
Kitaoka s theorem
to lattices over imaginary quadratic fields or quaternion division algebras.
Such tensor products provide several very good lattices. Bachoc and Nebe
[BacoN98] take
as
their starting point the lattice L
2
described on
p.
39
of
[Con16]. This provides a 10-dimensional representation over Z[a],
a
1 N / 2 for the group 2.M
22
.2. L
2
is an extremal 7-modular lattice
with minimal norm 8 and kissing number 6160.) Bachoc and Nebe form
the tensor product of
L
2
with ~ over Z[a] and obtain a 40-dimensional
extremal 3-modular lattice with minimal norm 8, and of L
2
with
8
to
obtain an SO-dimensional extremal unimodular lattice L
80
with minimal
norm 8 and kissing number 1250172000 (see Notes on Chapter
1).
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xi
viii
Preface to Third Edition
Lattices from Riemann surfaces
The period matrix
of
a compact Riemann surface
of
genus g determines
a real 2g-dimensional lattice. Buser and Sarnak [BuSa94] have shown
that
from
a sphere packing point of view these lattices are somewhat
disappointing: for large
g
their density is much worse than the Minkowski
bound, neither the root lattices E
6
,
E
8
nor the Leech lattice
can be
obtained, and so
on.
Nevertheless, for small genus some interesting lattices
occur [BerSI97], [Quin95], [QuZh95], [RiRo92], [Sar95], [TrTr84].
One example, the m.c.c. lattice, has already been mentioned in the
Notes
on
Chapter
I
The period matrix of the Bring curve (the genus
4 surface with largest automorphism group) was computed by Riera and
Rodriguez, and from this one can determine that the corresponding lattice
is
an
8-dimensional lattice
with
determinant
1,
minimal norm 1.4934
and kissing number 20 (see [NeSl]).
Lattices and codes wit no group
Etsuko Bannai [Bann90] showed that the fraction
of
n-dimensional
unimodular lattices with trivial automorphism group approaches 1
as
n --+
oo. Some explicit examples were given by Mimura [Mimu90]. Bacher
[Bace94] has found a
Type
I lattice
in
dimension
29
and a Type II lattice
in dimension 32 with trivial groups {1). Both dimensions are the lowest
possible.
Concerning codes, Ore and Phelps [OrPh92] proved that the fraction
of
binary self-dual codes
of
length n with trivial group approaches 1
as
n
--+ oo. A self-dual code with trivial group
of
length
34
(conjectured
to
be the smallest possible length) is constructed in [CoSl90a], and a doubly
even self-dual code
of
length
40
(the smallest possible)
in
[Ton89]. See
also [BuTo90], [Hara96], [Huff98], [LePR93], and [Leo8] (for a ternary
example).
Notes
on
Chapter
:
ounds
for
Codes and Sphere Packings
Samorodnitsky [Samo98] shows that the Delsarte linear programming
bound for binary codes is at least as large as the average of the Gilbert
Varshamov lower bound and the McEliece-Rodemich-Rumsey-Welch upper
bound, and conjectures that this estimate is actually the true value
of
the
pure linear programming bound.
K.rasikov
and Litsyn [KrLi97a] improve
on
Tietavainen s bound for
codes with
n 2
d
=
o n
3
.
Laihonen and Litsyn [LaiL98] derive a straight-line upper bound on
the minimal distance
of
nonbinary codes which improves
on
the Hamming,
linear programming and Aaltonen bounds.
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Preface
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Third Edition
xlix
Levenshtein [Lev87], [Lev91], [Lev92] and Fazekas and Levenshtein
[FaL95] have obtained new bounds for codes in finite and infinite
polynomial association schemes (cf. p. 247
of
Chap. 9).
Table
9.1
has been revised to include several new bounds on
A n, d).
A table of lower bounds on A n, d) extending t o n ~ 28 (cf. Table
9.1
of
Chap. 9) has been published by Brouwer et a . [BrSSS] (see also [Lits98]).
The main purpose
of
[BrSSS], however,
is
to present a table
of
lower
bounds on A n, d,
w)
for n ~ 28 (cf. 3.4 of Chap. 9).
Notes on hapter 1 : Three Lectures on Exceptional Groups
Curtis [Cur89a], [Cur90] discusses further ways to generate the Mathieu
groups M
12
and M
24
(cf. Chaps. 10,
11).
Hasan [Has89] has determined the possible numbers of common octads
in two Steiner systems S(5, 8, 24) (cf.
2.1 of
Chap. 10). The analogous
results for S(5, 6, 12) were determined by Kramer and Mesner in [KrM74].
Figure
10.1 of
Chap.
10
classifies the binary vectors
of
length 24
into orbits under the Mathieu group
24
[CoS190b] generalizes this
in
the following way. Let C be a code
of
length n over a field JF with
automorphism group
G,
and let
C.
denote the subset
of
codewords
of
C of
weight
w
Then
we
wish
to
classify the vectors
of lF"
into orbits
under
G,
and to determine their distances from the various subcodes
C. .
[CoS190b] does this for the first-order Reed-Muller, Nordstrom-Robinson
and Hamming codes of length 16, the Golay and shortened Golay codes
of
lengths 22, 23, 24, and the ternary Golay code
of
length
12.
For recent work
on
the subgroup structure
of
various finite groups
(cf. Postscript to Chap.
10)
see Kleidman et
a .
[KIL88), [KIPW89],
[KIW87), [KIW90), [KIW90a], Leibeck et a . [LPS90), Linton and Wilson
[LiW91), Norton and Wilson [NoW89], Wilson [Wi188], [Wi189]. The
modular version of the ATLAS of finite groups [Con16] has now
appeared [JaLPW).
On page 289,
in
the proof
of
Theorem 20, change
x
y
=
y
.
y
=
64
to x
x = y y =
64 . On page 292, 8th line from the bottom, change
{i)
to
{j)
Borcherds points out that
in
Table 10.4 on page
291
there
is
a third
orbit of type 10 vectors, with group M
.2. (Note that [Con16], p. 181,
classifies the vectors
up
to type 16).
Notes on hapter 11: The Golay odes
and
the Mathieu
Groups
For
more about the MOG (cf. 5 of Chap. 11) see Curtis [Cur89].
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rd Edition
The c
ohomology
o the groups
M
M
2
and
1
1
has been
studied i
n
[BenCa87],
[ AdM91] an
d [Cha82], re
spectively.
N
otes on Ch
apter
3:
B
ounds
on Kissing Nu
mbers
Drisch
and Sonneb
orn [DrS96] h
ave given
an upper bound
on the degr
ee
o
the
best polynom
ial to use
in the main the
orem
o
1
o Chap. 13.
Notes on C
hapter
5: On the Cl
assification
o
Integra
l
Quadratic Forms
A
recent book b
y Buell [Bue8
9] is
devoted
to the study
o binary qua
dratic
for
ms (cf. 3 o
Chap. 15 .
See also K
itaoka's book
[Kita93]
on the
arithmetic
theory
o
q
uadratic form
s (mentioned
already
in the Notes
on C
hapter 8
.
H
sia and Icaz
a [Hsic97] g
ive an effec
tive version
o
Tartako
vsky's theore
m.
For
an interpretatio
n
o
the "od
dity"
o
a la
ttice, see
Eq. (4) o the
Notes on C
hapter 4.
Tables
Nipp ([N
ip91] has con
structed a tab
le o
reduced
positive-def
inite integer
va
lued four-dim
ensional quad
ratic forms
o discriminant
::: 1732. A s
equel
[Nip9la]
tabulates five
-dimensional
forms o disc
riminant ::: 2
56. These
tab
les, together
with a new v
ersion
o
the
Brandt-Intrau
[Bra ] table
s o
ternary fo
rms computed
by Schiema
nn can also
be found on th
e electronic
atalogue of L
attices [NeSI].
Universal
forms
T
he
15-theorem
.
Conway
and Schneebe
rger [Schnee9
7], [CoSch98
] (see
al
so [CoFu97])
have shown
that for a pos
itive-definite q
uadratic form
with
int
eger matrix
entries
to
repr
esent all posi
tive integers
it suffices th
at it
represent
the numbers
I, 2, 3,
5,
6,
7, 10, 14
,
1
5. t
is conjectured
(the
290-con
jecture
that
for a positive
-definite quad
ratic form w
ith integer
value
s to represen
t all positive
integers it
suffices that
it represent
the
numb
ers I, 2,
3,
5
, 6, 7, 10,
13, 14, 15,
17, 19, 21, 2
2, 23, 26, 2
9,
30, 31
, 34, 35, 37,
42, 58, 93,
110, 145, 203
, 290.
T
he 15-theore
m is best-pos
sible in the
sense that
for each
o the
nine critica
l numbers
c
t
here is a po
sitive-definite
diagonal form
in four
variables tha
t misses only
c
For exam
ple
w
3x
2
4
5z
2
misses
only I, and
w
x
5
y
5
z
mis
ses only 15.
For the oth
er c in
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Preface to Third Edition
l i
the 290-conjecture the
forms are
not diagonal and sometimes involve
five
variables. For example a form that misses only 290
is
~
0
h
I
0
0
29
J lh
0
l
0 .
1416
29
For other work
on
universal forms see Chan, Kim and Raghavan [ChaKR],
Earnest and Khosravani [EaK97], [EaK97b], Kaplansky [Kap195].
M. Newman [Newm94] shows that any symmetric matrix
A
o
determinant d over a principal ideal ring
is
congruent to a tridigonal
matrix
[
d
0 0
dt c2 d2
0
0
d2
CJ
d3
dn 2 Cn 1
~ ~ ]
0
dn 1
Cn
in which
d
divides
d
for I :::; i ; 2. In particular, the Gram matrix for
a unimodular lattice can be put into tridiagonal form where all off-diagonal
entries except perhaps the last one are
equal
to
1.
[CSLDLl] extends the classification
o
positive definite integral lattices
o small determinant begun in Tables 15.8
and
15.9 o Chap. 15. Lattices
o determinants 4 and 5 are classified
in
dimensions n :::; 12 o determinant
6 in dimensions n :S 11 and o determinant up to 25 in dimensions n :::; 7.
The four 17-dimensional even lattices o determinant 2 (cf. Table 15.8)
were independently enumerated by Urabe [Ura89],
in
connection with
the classification o singular points on algebraic varieties. We note
that
in
1984 Borcherds [Borl, Table 2] had already classified the
121 25-dimensional
even
lattices o determinant 2. Even lattices o
dimension
16
and determinant 5 have been enumerated by Jin-Gen
Yang
[Yan94], and other lattice enumerations
in
connection with classification
o singularities can be found in [Tan91], [Ura87], [Ura90], [Wan91]. For
the connections between lattices
and
singularities, see Eberling [Ebe87],
Kluitmann [Klu89], Slodowy [Slod80], Urabe [Ura93], Voigt [Voi85].
Kervaire [Kerv94] has completed work begun by
Koch
and Venkov and
has shown that there
are
precisely 132 indecomposable even unimodular
lattices in imension 32 which have a complete root system (i.e. the
roots span the space). Only 119 distinct root systems occur.
Several recent papers have dealt with the construction and classifi
cations
o lattices, especially unimodular lattices, over rings o integers
in number fields, etc. See
for
example Bayer-Fluckiger
and
Fainsilber
[BayFa96], Benham et al. [BenEHH], Hoffman [Hof91], Hsia [Hsia89],
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Iii
Preface to Third Edition
Hsia and Hung [HsH89], Hung [Hun91], Takada [Tak85], Scharlau
[Scha94], Zhu [Zhu91HZhu95b].
Some related papers
on
class numbers
o
quadratic forms are
Earnest [Earn88HEarn91], Earnest and Hsia [EaH91], Gerstein [Gers72],
Hashimoto and Koseki [HaK86].
Hsia, Jochner and Shao [HJS], extending earlier work
o
Friedland
[Fri89], have shown that for any two lattices A and M
o
dimension
>
2
and in the same genus (cf. 7 o Chap. 15), there exist isometric primitive
sublattices
A
and M o codimension
I
Frohlich
and
Thiran [FrTh94] use the classification o Type I lattices
in studying the quantum Hall effect.
Erdos numbers
An old problem in combinatorial geometry asks how to place a given
number
o
distinct points
in
n-dimensional Euclidean space so
as
to
minimize the total number
o
distances they determine ([Chu84], [Erd46],
[ErGH89], [SkSL]). In 1946 Erdos [Erd46] considered configurations
formed by taking all the points
o
a suitable lattice that lie within a large
region. The best lattices for this purpose are those that minimize what we
shall call the
Erdos number
o the lattice, given by
=
Fdlln,
where d is the determinant o the lattice and F, its population fraction,
is given by
F
_
1
P(x)
1m . i n
2:::3,
X--+00 X
where
P(x) is
the population function
o
the corresponding quadratic form,
i.e. the number o values not exceeding
x
taken by the form.
2
The Erdos
number
is
the population fraction when the lattice
is
normalized to have
determinant I.
It
turns out that minimizing E is an interesting problem
in pure number theory.
In
[CoS191]]
we
prove all cases except n
=
2 (handled by Smith
[Smi91]) o the following proposition:
The lattices with minimal Erdos number are up to a scale factor)
the even lattices o minimal determinant. For n = 0, 1, 2, . . . these
determinants are
1, 2, 3, 4, 4, 4, 3, 2, 1, 2, 3, 4, 4, 4,
this sequence continuing with period 8.
2
For n
::
2 these definitions must be modified. For n
=
0 and 1
we
set = 1, while for n =2 we define F by F = imx ...oox-
1
P(x)J ogx.
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Pref
ace to
Third E
dition
For n
::::
1
0 these
lattice
s are u
nique:
with
Erdos
numb
ers
(
1
-112
1 1,
T
312
3
1
/
4
n
1 -
2
=
0.5533
,
p=2(3
)
41/4
. .. .4
1
3
=
0.7276
,
2
4
2
= 0.7071
,
41/5
2 = 0.6
598,
21/7
2
=
0
.5520,
2
1/9
2
2 =0.540
0,
31/6
2 =0.600
5,
3
1/10
-2 -
= 0.5
581,
iii
(roun
ded to
4 dec
imal p
laces),
while
for ea
ch n
t
here a
re two
or
mo
re suc
h lattic
es. T
he pro
of use
s the
p-adic
structu
res
o the l
attices
( c
f. Cha
p. 15).
The
three-
dimens
ional c
ase
is the m
ost d
ifficult.
The
crucial
numbe
r-theor
etic re
sult nee
ded fo
r our p
roof was
firs
t estab
lished b
y
Pet
ers [Pe
t80] u
sing th
e Gene
ralized
Riema
nn Hy
pothesi
s. The
depen
dence
on t
his hyp
othesis
has b
een rem
oved
by Du
ke and
Schul
ze-Pillo
t [DuS
90].
Notes
on
hapter
6:
Enumeration
o
Unimodular Lattices
Recent
work
on th
e clas
sificatio
n o
lattices
o
va
rious
types h
as be
en
describ
ed in
the No
tes on
Chapt
er
15.
The
m
ss for
mul
o H. J.
S. Smit
h, H.
M
inkow
ski an
d C.
L Sieg
el
(c
f. 2
o Ch
ap. 1
6) expre
sses th
e sum
o
th
e recip
rocals
o the
group
order
s o th
e lattic
es in
a genu
s in te
rms
o the p
ropertie
so a
ny one
o
them
. In [C
SLDL
4] we
discuss
the hi
story
o the f
ormula
and r
estate i
t in
a
way th
at mak
es it e
asier to
comp
ute.
In partic
ular we
give
a simp
le and
re
liable w
ay to
evalua
te the
2-adic
contrib
ution.
Our ve
rsion,
unlike
earlier
ones,
is
visibly invariant under scale changes and dualizing.
We
then
use th
e form
ula to
check
the en
umera
tion
o lattice
s
o
d
etermin
ant d
::::
25
given
in [C
SLDLI
]. [CS
LDL4]
also
contain
s table
s
o
t
he sta
ndard
mas
s, val
ues
o the L
-series
I; f,;)m
-s m
odd),
and ge
nera
o lattic
es
o
dete
rminan
t d
:::: 25.
Eskin,
Rudn
ick an
d Sarn
ak
give a
new p
roof
o the
mass
f
ormula
using
an
orbit-co
unting
meth
od. A
nothe
r proo
f is g
iven b
y
Mis
chler [
Misch9
4].
T
he cl
assifica
tion o
the
Niemei
er latt
ices
is reder
