The review by Gian-Carlo Rota, in Advances in Mathematics, Vol. 84, Number 1, Nov. 1990, p. 136, reads, in full: ``This is the best survey of the best work in one of the best fields of combinatorics, written by the best people. It will make the best reading by the best students interested in the best mathematics that is now going on.''
The review by G. David Forney, Jr., in the IEEE Trans. Information Theory, Vol. 36, July 1990, pp. 955-956 concludes with: ``What is so often said in book reviews happens to be precisely true here: this book will be an essential reference for anyone whose work involves lattices for the forseeable future. There is nothing else like it, and as an intellectual accomplishment it is breathtaking.''
ISBN: 0-387-98585-9
Series: Grundlehren der mathematischen Wissenshaften, Volume 290;
lxiv + 703 pp.
Springer-Verlag,
(800) 777 4643 or (201) 348 4033;
44 Hartz Way, Secaucus, NJ 07096.
Should be ordered from the publisher, Springer-Verlag, or from Amazon.com, etc.
At the present time $Q_{33}$ and $Q_{34}$ are the densest packings known in those dimensions, and $Q_{32} , \ldots, Q_{40}$ have the highest kissing numbers presently known for {\em lattice} packings. The lattices $Q_{30}$ and $Q_{31}$ will be found on page 220. The lattice $P_{48n}$ mentioned at dimension 48 is the extremal lattice found by Nebe \cite{Nebe98}.
A new table of kissing numbers. Table I.2 gives the highest kissing numbers presently known in dimensions $n \le 128$. If a dimension $n$ is not mentioned in the appropriate column, let $m$ be the next lowest dimension that is mentioned, and use the sum of the entries for $m$ and $n-m$. For more information about most of these packings see Tables 1.2 and 1.3. (Parenthesized entries indicate that higher kissing numbers can be obtained from nonlattice packings.)
The kissing number of the Mordell-Weil lattice $MW_{44}$ was computed by G. Nebe (personal communication), and that of $MW_{128}$ by Elkies \cite{Elki}. However, a simple construction using binary codes yields higher kissing numbers \cite{EdRS98}.
A. Vardy has pointed out to us that by using the Nordstrom-Robinson code as inner code, and the Vladuts-Katsman-Tsfasman algebraic-geometry codes as outer codes, as in \cite{TsV91}, Theorem 3.4.16, one obtains a polynomial-time construction for a family of nonlinear binary codes with $d/n \ge 0.25$ and rate $R= k/n = 2/15 \: (1+ o(1))$ as $n \to \infty$. Thus there is a polynomial-time construction for nonlattice packings with kissing number\index{kissing number}
a considerable improvement over Eq. (56) of Chap. 2. No polynomial-time construction is presently known however for a sequence of lattices in which the kissing number grows exponentially with dimension. See also \cite{Alon97}.
\cite{CSLDL3} contains a simple and self-contained proof of the classification of {\bf perfect lattices}\index{perfect lattices} in dimensions $n \le 4$ and hence of the determination of the densest lattice packings in these dimensions (cf. Table~1.1 of Chap.~1). The main goal of \cite{CSLDL3} is to study the perfect lattices in dimensions $n \le 7$ found by Korkine and Zolotareff [Kor3], Voronoi [Vor1], Barnes [Bar6] -- [Bar9], Scott [Sco1], [Sco2], Stacey\index{Stacey} [Sta1], [Sta2], and others, and to determine their automorphism groups, orbits of minimal vectors, and eutactic coefficients. It is shown that just 30 of the 33 seven-dimensional perfect lattices are extreme. Jaquet\index{Jacquet} \cite{Jaq93} has now shown that this list of 33 seven-dimensional perfect lattices is complete. See also \cite{Anz91}, \cite{BatMa94} and especially Martinet\index{Martinet} \cite{Mar96}.
"If P is a lattice then the vertices of the Voronoi cells are exactly the holes, but in general they may include other points".
This is wrong! It should say:
"Holes are necessarily vertices of Voronoi cells, but vertices of Voronoi cells need not be holes (even if one only considers lattices)."
[Thanks to Gabriele Nebe for telling us about this mistake, and to Frank Vallentin who found a counterexample, a three-dimensional lattice for which not all the vertices of the Voronoi cells are holes.]
Alex Healy (ahealy@fas.harvard.edu) points out that the last three sentences on this page need to be corrected. He says:
The problem of computing the covering radius of a lattice is not known to be NP-hard, although determining the packing radius is known to be NP-hard (Ajtai, 1998). In fact, it is known that approximating the packing radius to within a factor of sqrt(2) is NP-hard as well (Micciancio, 1998). The cited paper, "[Emd1]", proves that the closest vector problem in the Euclidean and L-infinity norms is NP-hard, and conjectures that the shortest vector problem (essentially the packing radius problem) is NP-hard. The analogous problems for binary codes (minimum weight codeword, nearest codeword, covering radius) are all known to be NP-hard [Ber8], [McL2].
Additional references:
1. Michael Tsfasman (tsfasman@iitp.ru) points out that there is an error on page 225. Omit the second inequality in Equation (55b) and the sentence containing Eq. (56), but still define c to be 1 if K is totally real and 1/2 if K is totally complex. Assume K is totally real up to the beginning of Theorem 8. Then Theorem 8 still holds (for both cases).
2. Maurice Craig has also pointed out that are problems with these pages. See his document: Maurice Craig, Minkowski's Embedding. A cached copy of this document is also available here.
2(+- a_{rs} +- a_{rt} +- a_{st} ≤ a_{rr} + a_{ss} (r<s<t) where the three +- signs have product -1.
(Thanks to Jerry Shurman of Reed College and David Yuen of Lake Forest College for correcting an earlier "correction" of this equation.)