WHATIS... ? the Leech Lattice? Chuanming Zong

The Leech lattice is a magical structure in twenty- the final entry, 0, first, followed by the other 24 four-dimensional Euclidean space E that was 22 entries, we get the first cyclic shift of w1: inspired by Golay’s error-correcting code G24. The (0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). magic of the Leech lattice led Conway to the dis- The next cyclic shift is (0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, covery of the three sporadic simple groups: Co1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), and so on. Then w1 and its Co2, and Co3. Also magically, the Leech lattice cyclic shifts generate a binary code, the Golay code provides the optimal kissing configuration for the G23, which was discovered by M. J. E. Golay in 1949. 24-dimensional unit ball as well as the densest The minimal Hamming distance of this code is lattice ball packing in E24. seven, and therefore it can detect and correct three- Data in digital systems are typically stored, trans- bit errors. This code has 212 = 4096 codewords. mitted, and processed in binary codewords. If a sin- By adding a parity check to each codeword of G23, gle codeword is in error, the message is garbled or we get the extended Golay code G24. The minimal the computation spoiled. Starting in the 1940s, scien- Hamming distance of G24 is eight. tists searched for coding systems that could detect The philosophy of error-correcting codes—to de- and even correct errors. sign codes with both large minimal Hamming dis- tances and large numbers of codewords—is related message to ball packings with large packing densities. In 1965 M J. Leech constructed a twenty-four-dimensional lat- tice Λ by lifting the extended Golay code G24 from 24 24 Z2 to Z and restricting the sum of the coordinates encoder channel decoder to zero modulo 4. Here an n-dimensional lattice is the set of all linear combinations of n linearly inde- pendent vectors over Z. In 1967 Leech realized that message there are big holes in Λ. Filling those holes doubles M’ the density and produces a remarkable lattice, Λ24, Figure 1. The data transmission process. the Leech lattice. For convenience, we say a vector j k (v1, v2, . . . , vn) has shape (a , b , . . .) if vi = a for j In 1947 R. Hamming discovered the first binary entries, vi = b for k entries, etc. In fact, the Leech lattice can be generated by all vectors of the shape error-correcting code, H7, which is generated by four vectors, (1, 1, 0, 1, 0, 0, 0), (0, 1, 1, 0, 1, 0, 0), √1 (∓3, ±123), 8 (0, 0, 1, 1, 0, 1, 0) and (0, 0, 0, 1, 1, 0, 1), over Z2. The Hamming distance between two codewords is the where the ∓3 can be in any position and the up- number of their different entries. The minimal per signs are taken on a set of coordinates where Hamming distance of the code above is four, and a codeword of G24 is one. therefore this code can detect and correct single-bit The Leech lattice has 196,560 shortest vectors of 16 8 errors. length two: 97,152 of them have shape (0 , ±2 ); 98,304 of them have shape (±123, ±3); and 1,104 Let w1 denote the binary 23-tuple (1, 1, 0, 0, 0, 1, 22 2 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). If we write of them have shape (0 , ±4 ). One might therefore conjecture that Λ24 has a large symmetry group. In 1968 J. H. Conway determined this group, Co . It is Chuanming Zong is professor of mathematics at Peking Uni- 0 versity. His email address is [email protected]. generated by six elements and has order 22 9 4 2 DOI: http://dx.doi.org/10.1090/noti1045 |Co0| = 2 3 5 7 11 · 13 · 23.

1168 Notices of the AMS Volume 60, Number 9 More surprisingly, he discovered three new spo- and E24. Perhaps the mystery lurking in the back- 1 radic simple groups, Co1, Co2, and Co3, as ground is the uniqueness of the optimal configura- subgroups of Co0, where tions. Let δ(Bn) and δ∗(Bn) denote the densities of the |Co | = 22139547211 · 13 · 23, 1 densest packings and the densest lattice packings of 18 6 3 n |Co2| = 2 3 5 7 · 11 · 23, B respectively. It can be verified that the determi- and nant of Λ24 is one and therefore the packing density 10 7 3 24 + 12 |Co3| = 2 3 5 7 · 11 · 23. of B Λ24 is π /12!. Thus we have Let Bn denote the n-dimensional unit ball π 12 δ(B24) ≥ δ∗(B24) ≥ . centered at the origin, that is, 12! n Bn = {x ∈ En : |x| ≤ 1}; For a real function f (x) defined on E we define Z let τ(Bn) denote its kissing number (the maximal fb (y) = f (x)e2πihy,xidx, n number of nonoverlapping unit balls that can simul- E taneously touch Bn at its boundary); and let τ∗(Bn) where√ hy, xi is the inner product of y and x, and i = denote its lattice kissing number. Since the length of −1. If there is a positive constant µ such that both 24 | | | | the shortest vectors in 24 is two, B + 24 is a lattice f (x) and fb (x) are bounded above by a constant Λ Λ −n−µ ball packing. Therefore we have times (1 + |x|) , we say f (x) is admissible. In 2003 H. Cohn and N. D. Elkies proved the fol- (1) τ(B24) ≥ τ∗(B24) ≥ 196560. lowing criterion: Suppose f (x) is an admissible func- Let A(n, θ) denote the maximal number of points tion defined on En that satisfies: n on the surface of B with minimal spherical separa- 1. f (o) = fb (o), tion θ. Clearly we have 2. f (x) ≤ 0 whenever |x| ≥ r, and n τ(Bn) = A(n, π/3). 3. fb (x) ≥ 0 for all x ∈ E . Then we have For k = 0, 1, 2,..., let P α,β(t) denote the Jacobi k π n/2  r n polynomial of degree k, where α > −1 and β > −1 δ(Bn) ≤ . (n/2)! 2 are two parameters. These polynomials form an or- thogonal basis for the space of all polynomials. In Actually, this is a Euclidean analog of (2). Based the 1970s P. Delsarte et al. discovered the following on this result, in 2009 H. Cohn and A. Kumar proved criterion: Write α = (n − 3)/2. If π 12   π 12 (4) ≤ δ(B24) ≤ 1 + 1.65 · 10−30 · k 12! 12! X α,α f (t) = fiPi (t) and 12 i=0 π δ∗(B24) = . is a real polynomial such that f0 > 0, fi ≥ 0 for i = 1, 12! 2, . . . , k, and f (t) ≤ 0 for −1 ≤ t ≤ cos θ, then This time, up to symmetry, the Leech lattice again is the only optimal 24-dimensional lattice for δ∗(B24). f (1) (2) A(n, θ) ≤ . Doubtless, given (4), everybody will bet on f0 π 12 In 1978 V. I. Levenštein, A. M. Odlyzko, and N. J. A. δ(B24) = . 12! Sloane constructed such a polynomial f (t) for n = 24 and surprisingly obtained Acknowledgments (3) τ(B24) = A(24, π/3) ≤ 196560. This work is supported by 973 Programs 2013CB83- 4201 and 2011CB302401, the National Natural Sci- Then (1) and (3) together yield ence Foundation of China (grant No. 11071003), and 24 ∗ 24 τ(B ) = τ (B ) = 196560. the Chang Jiang Scholars Program of China. I am Moreover, as was shown by E. Bannai and N. J. A. grateful to the referee for helpful comments. Sloane in 1981, the local kissing configuration of 24 24 References B + Λ24 is the only optimal one for τ(B ), up to isometry. 1. H. Cohn and A. Kumar, Optimality and uniqueness The problem of optimizing the upper bound in of the Leech lattice among lattices, Ann. Math. (2) 170 (2) is unsolved in general and appears to be difficult. (2009), 1003–1050. 2. J. H. Conway and N. J. A. Sloane, Sphere Packings, However, there are simple choices of f (t) that ex- 8 Lattices and Groups, Springer-Verlag, New York, 1988; actly solve the kissing number problem in both E 3rd ed, 1999. 3. T. M. Thompson, From Error Correcting Codes through 1 There are twenty-six sporadic simple groups. The first was Sphere Packings to Simple Groups, Mathematical discovered in 1861 by E. Mathieu, and the last one, known Association of America, 1983. as the friendly giant or the monster, was constructed by 4. C. Zong, Sphere Packings, Springer-Verlag, New York, R. Griess in 1982 (see “What is the monster?”, by Richard 1999. Borcherds, Notices, October 2002).

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