Bulletin of the American Mathematical Society

MathSciNet review: 1567781
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Book Information:

Authors: J. H. Conway and N. J. A. Sloane
Title: Sphere packings, lattices and groups
Additional book information: Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1988, xxviii + 663 pp., $87.00. ISBN 0-387-96617-X, and ISBN 3-540-96617-X.

Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canadian J. Math. 33 (1981), no. 2, 437–449. MR 617634, DOI 10.4153/CJM-1981-038-7

2.

C. Bender, Bestimmung der grössten Anzahl gleich Kugeln, welche sich auf ein Kugel von demselben Radius, wie die übrigen, auflegen lassen, Arch Math. Phys. (Grunert) 56 (1874), 302-306.

3.

Elwyn R. Berlekamp, John H. Conway and Richard K. Guy, Winning Ways for your Mathematical Plays, Chapter 14, Academic Press, New York, 1982.

A. H. Boerdijk, Some remarks concerning close-packing of equal spheres, Philips Research Rep. 7 (1952), 303–313. MR 50302

R. E. Borcherds, J. H. Conway, L. Queen, and N. J. A. Sloane, A monster Lie algebra?, Adv. in Math. 53 (1984), no. 1, 75–79. MR 748897, DOI 10.1016/0001-8708(84)90018-5

J. H. Conway, A characterisation of Leech’s lattice, Invent. Math. 7 (1969), 137–142. MR 245518, DOI 10.1007/BF01389796

J. H. Conway, Three lectures on exceptional groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 215–247. MR 0338152

J. H. Conway, The automorphism group of the $26$-dimensional even unimodular Lorentzian lattice, J. Algebra 80 (1983), no. 1, 159–163. MR 690711, DOI 10.1016/0021-8693(83)90025-X

J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI 10.1112/blms/11.3.308

J. H. Conway, A. M. Odlyzko, and N. J. A. Sloane, Extremal self-dual lattices exist only in dimensions $1$ to $8$, $12$, $14$, $15$, $23$, and $24$, Mathematika 25 (1978), no. 1, 36–43. MR 505767, DOI 10.1112/S0025579300009244

S. Norton, A bound for the covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1779, 259–260. MR 660414, DOI 10.1098/rspa.1982.0041

J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15 (1982), no. 1, 83–94. MR 666350, DOI 10.1016/0022-314X(82)90084-1

J. H. Conway and N. J. A. Sloane, Voronoĭ regions of lattices, second moments of polytopes, and quantization, IEEE Trans. Inform. Theory 28 (1982), no. 2, 211–226. MR 651816, DOI 10.1109/TIT.1982.1056483

J. H. Conway and N. J. A. Sloane, Twenty-three constructions for the Leech lattice, Proc. Roy. Soc. London Ser. A 381 (1982), no. 1781, 275–283. MR 661720, DOI 10.1098/rspa.1982.0071

J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 215–217. MR 640949, DOI 10.1090/S0273-0979-1982-14985-0

J. H. Conway and N. J. A. Sloane, Laminated lattices, Ann. of Math. (2) 116 (1982), no. 3, 593–620. MR 678483, DOI 10.2307/2007025

J. H. Conway and N. J. A. Sloane, Leech roots and Vinberg groups, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 233–258. MR 684311, DOI 10.1098/rspa.1982.0157

J. H. Conway and N. J. A. Sloane, The unimodular lattices of dimension up to $23$ and the Minkowski-Siegel mass constants, European J. Combin. 3 (1982), no. 3, 219–231. MR 679207, DOI 10.1016/S0195-6698(82)80034-6

John H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Trans. Inform. Theory 32 (1986), no. 3, 337–348. MR 838197, DOI 10.1109/TIT.1986.1057187

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. III. Perfect forms, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 43–80. MR 953277

21.

C. F. Gauss, Besprechung des Buchs von L. A. Seeber: Untersuchungen über die Eigenschaften der positiven ternaren quadratischen Formen usw., Göttingsche gelehrte Anzeigen (1831-07-09)=Werke, II (1876), 188-196.

22.

S. Günther, Ein stereometrisches Problem, Arch. Math. Phys. (Grunert) 57 (1875), 209-215.

23.

R. Hoppe, Bemerkung der Redaktion, Arch. Math. Phys. (Grunert) 56 (1874), 307-312.

John Leech, The problem of the thirteen spheres, Math. Gaz. 40 (1956), 22–23. MR 76369, DOI 10.2307/3610264

John Leech and N. J. A. Sloane, Sphere packings and error-correcting codes, Canadian J. Math. 23 (1971), 718–745. MR 285994, DOI 10.4153/CJM-1971-081-3

James G. Propp, Kepler’s spheres and Rubik’s cube, Math. Mag. 61 (1988), no. 4, 231–239. MR 962584, DOI 10.2307/2689358

S. Norton, A bound for the covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1779, 259–260. MR 660414, DOI 10.1098/rspa.1982.0041

A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in $n$ dimensions, J. Combin. Theory Ser. A 26 (1979), no. 2, 210–214. MR 530296, DOI 10.1016/0097-3165(79)90074-8

Arnold Pizer, A note on a conjecture of Hecke, Pacific J. Math. 79 (1978), no. 2, 541–548. MR 531334

C. A. Rogers, The packing of equal spheres, Proc. London Math. Soc. (3) 8 (1958), 609–620. MR 102052, DOI 10.1112/plms/s3-8.4.609

N. J. A. Sloane, Recent bounds for codes, sphere packings and related problems obtained by linear programming and other methods, Papers in algebra, analysis and statistics (Hobart, 1981) Contemp. Math., vol. 9, Amer. Math. Soc., Providence, R.I., 1981, pp. 153–185. MR 655979

Thomas M. Thompson, From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, Mathematical Association of America, Washington, DC, 1983. MR 749038

B. B. Venkov, On the classification of integral even unimodular $24$-dimensional quadratic forms, Trudy Mat. Inst. Steklov. 148 (1978), 65–76, 273 (Russian). Algebra, number theory and their applications. MR 558941

1.

Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), 437-449. MR 0617634

2.

C. Bender, Bestimmung der grössten Anzahl gleich Kugeln, welche sich auf ein Kugel von demselben Radius, wie die übrigen, auflegen lassen, Arch Math. Phys. (Grunert) 56 (1874), 302-306.

3.

Elwyn R. Berlekamp, John H. Conway and Richard K. Guy, Winning Ways for your Mathematical Plays, Chapter 14, Academic Press, New York, 1982.

4.

A. H. Boerdijk, Some remarks concerning close-packing of equal spheres, Philips Res. Rep. 7(1952), 303-313. MR 50302

5.

R. E. Borcherds, J. H. Conway, L. Queen and N. J. A. Sloane, A monster Lie algebra? Adv. in Math. 53 (1984), 75-79. MR 748897

6.

J. H. Conway, A characterization of Leech's lattice, Invent. Math. 7 (1969), 137-142. MR 245518

7.

J. H. Conway, Three lectures on exceptional groups, Finite Simple Groups (M. B. Powell and G. Higman, eds. ), Academic Press, New York, 1971, pp. 215-247. MR 338152

8.

J. H. Conway, The automorphism group of the 26-dimensional even unimodular Lorentzian lattice, J. Algebra 80 (1983), 159-163. MR 690711

9.

J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339. MR 554399

10.

J. H. Conway, A. M. Odlyzko and N. J. A. Sloane, Extremal self-dual lattices exist only in dimensions 1 to 8, 12, 14, 15, 23 and 24, Mathematika 25 (1978), 36-43. MR 505767

11.

J. H. Conway, R. A. Parker and N. J. A. Sloane, The covering radius of the Leech lattice, Proc. Roy Soc. London A380 (1982), 261-290. MR 660415

12.

J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 14 (1982), 83-94. MR 666350

13.

J. H. Conway and N. J. A. Sloane, Voronoi regions of lattices, second moments of polytopes, and quantization, I.E.E.E. Trans. Info. Theory 28 (1982), 211-226. MR 651816

14.

J. H. Conway and N. J. A. Sloane, Twenty-three constructions for the Leech lattice, Proc. Roy. Soc. London A381 (1982), 275-283. MR 661720

15.

J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 215-217. MR 640949

16.

J. H. Conway and N. J. A. Sloane, Laminated lattices, Ann. of Math. (2) 116 (1982), 593-620. MR 678483

17.

J. H. Conway and N. J. A. Sloane, Leech roots and Vinberg groups, Proc. Roy. Soc. London A384 (1982), 233-258. MR 684311

18.

J. H. Conway and N. J. A. Sloane, The unimodular lattices of dimension up to 23 and the Minkowski-Siegel mass constants, Europ. J. Combin. 3 (1982), 219-231. MR 679207

19.

J. H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, I.E.E.E. Trans. Info. Theory 32 (1986), 337-348. MR 838197

20.

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. III Perfect forms, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 43-80. MR 953277

21.

C. F. Gauss, Besprechung des Buchs von L. A. Seeber: Untersuchungen über die Eigenschaften der positiven ternaren quadratischen Formen usw., Göttingsche gelehrte Anzeigen (1831-07-09)=Werke, II (1876), 188-196.

22.

S. Günther, Ein stereometrisches Problem, Arch. Math. Phys. (Grunert) 57 (1875), 209-215.

23.

R. Hoppe, Bemerkung der Redaktion, Arch. Math. Phys. (Grunert) 56 (1874), 307-312.

24.

J. Leech, The problem of the thirteen spheres, Math. Gaz. 40 (1956), 22-23. MR 76369

25.

J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Canad. J. Math. 23 (1971), 718-745. MR 285994

26.

James G. Propp, Kepler's spheres and Rubik's cube, Math. Mag. 61 (1988), 231-239. MR 962584

27.

S. P. Norton, A bound for the covering radius of the Leech lattice, Proc. Roy. Soc. London A380 (1982), 259-260. MR 660414

28.

A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), 210-214. MR 530296

29.

Arnold Pizer, A note on a conjecture of Hecke, Pacific J. Math. 79 (1978), 541-548; MR 80g: 10028. MR 531334

30.

C. A. Rogers, The packing of equal spheres, Proc. London Math. Soc. 8 (1958), 609-620. MR 102052

31.

N. J. A. Sloane, Recent bounds for codes, sphere packings and related problems obtained by linear programming and other methods, Contemporary Math. vol. 9, Amer. Math. Soc. Providence, R. I., 1982, pp. 153-185. MR 655979

32.

Thomas M. Thompson, From error-correcting codes through sphere packings to simple groups, Carus Math. Monograph 21, Math. Assoc. Amer., Washington DC, 1983. MR 749038

33.

B. B. Venkov, The classification of integral even unimodular 24-dimensional quadratic forms, Trudy Mat. Inst. Steklov 148 (1978), 65-76=Proc. Steklov Inst. Math. no. 4 (1980), 63-74. MR 558941