Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31
aa r X i v : . [ m a t h . M G ] A ug IMPROVED LOWER BOUNDS FOR KISSING NUMBERS INDIMENSIONS 25 THROUGH 31
KENZ KALLAL, TOMOKA KAN, AND ERIC WANG
Abstract.
The best previous lower bounds for kissing numbers in dimensions25–31 were constructed using a set S with | S | = 480 of minimal vectors of theLeech Lattice, Λ , such that h x, y i ≤ for any distinct x, y ∈ S . Then, a prob-abilistic argument based on applying automorphisms of Λ gives more disjointsets S i of minimal vectors of Λ with the same property. Cohn, Jiao, Kumar,and Torquato proved that these subsets give kissing configurations in dimen-sions 25–31 of given size linear in the sizes of the subsets. We achieve | S | = 488 by applying simulated annealing. We also improve the aforementioned proba-bilistic argument in the general case. Finally, we greedily construct even larger S i ’s given our S of size , giving increased lower bounds on kissing numbersin R through R . Introduction
Kissing Numbers. A kissing configuration in R n is a set of non-overlappingunit spheres externally tangent to the unit sphere S n − . The kissing number in R n is the maximal size of kissing configurations in R n . The kissing number is knownonly for dimensions 1, 2, 3, 4, 8 and 24 (the bold figures in Table 1). In dimensionsDimension Kissing Number Dimension Kissing Number1
13 11542
14 16063
15 25644
16 43205 40 17 53466 72 18 73987 126 19 106688
20 174009 306 21 2772010 500 22 4989611 582 23 9315012 840 24
Table 1.
The best lower bounds known for kissing numbers indimensions 1–24.25 to 31, the best lower bounds previously known for kissing numbers were provedin [CJKT11]. We improve these bounds by using computer programs to maximize
Date : March 27, 2018.2010
Mathematics Subject Classification. imension New Bounds Previous Bounds (2011)25 197048 19704026 198512 19848027 199976 19991228 204368 20418829 208272 20793030 219984 21900831 232874 230872 Table 2.
New kissing number lower bounds in dimensions 25–31.Dimension Lower Bounds on Kissing Number25 | S | | S | + 2 | S | | S | + 2 | S | + P i =3 | S i | P i =1 | S i | P i =1 | S i | + P i =9 | S i | P i =1 | S i | P i =1 | S i | + P i =25 | S i | Table 3.
Lower bounds on kissing numbers in R n the size of an initial set of minimal vectors of the Leech lattice, and to constructsubsequent mutually disjoint sets of minimal vectors. We also improve [CJKT11]’sprobabilistic argument for the construction of these subsets, which improves thelower bounds, despite being surpassed by the computer-aided construction in spe-cific cases. We summarize our results in Table 2. Although we set new recordsfor the lower bounds, our constructions are not optimal. However, we introducetechniques which may help improve this approach further.Next, we introduce some basic concepts and the main setup in [CJKT11] tohelp explain our new results in Sections 2–4. Basic geometric considerations allowus to reformulate the kissing number in R n as the maximum number N of points x , . . . , x N on the unit sphere S n − such that h x i , x j i ≤ / for i = j . Thisreformulation makes it significantly easier to test large kissing configurations.1.2. The Kissing Number in 24 Dimensions.
The highest-dimensional Eu-clidean space in which the kissing number is known is R , where the centers ofthe tangent spheres are the 196560 minimal vectors of the Leech lattice Λ (see[CS99]). The optimality of Λ was proven in [Lev79] and [OS79].Let C denote the set of minimal vectors of Λ . Every element of C is of the“shape” ( ± , ) / √ , ( ± , ) / √ , or ( ± , ± ) / √ , where ( ± a u , ± b w ) repre-sents vectors which contain u copies of ± a and w copies of ± b with signs independentof each other (see [Tho83]). Note that not every such vector is in C . There area total of |C| = 196560 minimal vectors, all with norm . Let S be a subset of C which satisfies h x, y i ≤ for all distinct x, y ∈ S . We will see that maximizing | S | will allow us to increase the lower bounds on kissing numbers in R through R .1.3. Dimensions past 24.
Given mutually disjoint subsets S i of C satisfying thesame inner product condition as S , we construct configurations in R d that yieldthe lower bounds in Table 3. These are a special case of [CJKT11, Theorem 7.5]: Theorem 1.1.
Let S , . . . , S n be mutually disjoint subsets of C with h x, y i ≤ forall distinct x, y ∈ S i , where i = 1 , , . . . , n . Suppose that we have partitioned a issing configuration in S d − into disjoint subsets T , . . . , T n such that for each i , h x, y i ≤ − / for any x, y ∈ T i . Then the kissing number in R d is at least n X i =1 ( | T i | − | S i | . Sketch of proof.
We can verify that the following set of vectors of norm has pair-wise dot product at most 2: ( ( x, ∈ R × R d | x ∈ C\ n [ i =1 S i ) ∪ n [ i =1 n(cid:16) x p / , y p / (cid:17) | x ∈ S i , y ∈ T i o . (cid:3) Note that | T i | ≤ , where equality holds when T i contains three vectors arrangedin an equilateral triangle in the same plane as the origin.[CJKT11] gives the method indicated in Proposition 1.3 below for obtaininglower bounds on the | S i | ’s probabilistically, which we later improve in Section 4.Their proof uses the automorphism group of Λ , defined below: Definition 1.2 (Automorphism group of Λ ) . The automorphisms of Λ are thebijections g : Λ → Λ which preserve distance in R and fix the origin. Theseautomorphisms form a group acting on Λ , known as the Conway group Co .It follows from this definition that automorphisms also preserve inner product.We present [CJKT11, Lemma 7.4] in Proposition 1.3. The proof of this lemmainvolves recursively bounding the | S i | ’s from Theorem 1.1 by translating S by arandom element g ∈ Co , and deleting the intersection of gS with S ∪ · · · ∪ S i − .By using the bounds on the | S i | ’s given in Proposition 1.3, the bounds in Table 3,and an initial S of size 480, [CJKT11] arrived at the numerical lower bounds onkissing numbers displayed in the rightmost column of Table 2. Proposition 1.3. [CJKT11]
Let S = S . Then for any n ≥ , there exist mutuallydisjoint subsets S , . . . , S n ⊂ C such that h x, y i ≤ for all x, y ∈ S i , and | S i | ≥ | S | − P i − j =1 | S j ||C| ! = | S | − P i − j =1 | S j | ! . Moreover, if S is antipodal, each of the S i ’s will also be antipodal. Starting from an antipodal set S with | S | = 480 , [CJKT11] applied Theorem 1.1and Proposition 1.3 to give the lower bounds shown in the rightmost column ofTable 2.These two results yield two clear methods to increase the lower bounds on kissingnumbers in R through R : constructing S of size larger than 480, and construct-ing S i ’s which improve on the probabilistic bounds given by Proposition 1.3. Weimprove the former in Section 2, and the latter in Sections 3 and 4.2. Constructing a First Subset
We constructed an S with | S | = 488 by computer methods outlined below. Thisset of vectors can be found in the text file listed in Appendix A.3 lgorithm 2.1 Greedy Construction of S procedure Greedy_S ( C ) S ← ∅ for all v in C do if h v , v i ≤ for all v ∈ S \ { v } then S ← S ∪ { v } end if end for return S end procedure Greedy Approach.
In order to generate their S with | S | = 480 , [CJKT11]used the greedy algorithm outlined in Algorithm 2.1. In order to iterate over theelements of C , the algorithm requires a certain ordering of the vectors. The averagesize of a set S generated from a random order seems to hover around 238. Algorithm 2.2
Simulated Annealing procedure Anneal_S ( C ) S ← ∅ for ≤ t ≤ t max do T ← temp ( t ) if rand (0 , < and ∃ v ∈ C \ S such that h v , x i ≤ ∀ x ∈ S then c ← random element of C \ S satisfying the above property S ← S ∪ { c } else s ← random element of S S ′ ← S \ { s } p ← exp( − ( energy ( S ′ ) − energy ( S )) /T ) if rand(0, 1) < p then S ← S ′ end if end if end for return S end procedure Simulated Annealing.
We improved on this approach by using the simulatedannealing algorithm shown in Algorithm 2.2, which minimizes the energy of thestate S , whose value depends on an arbitrary energy function. We used the functionenergy ( S ) = −| S | , but any function of the current state that tends to decrease with | S | could work. (For instance, a function that takes into consideration the numberof vectors in C \ S that can still be added to the configuration S ∪ { v } .)The temperature temp ( t ) describes how likely the algorithm is to move to a lessoptimal state, and is a function of the number of iterations. In a process called the cooling schedule , the algorithm starts at a high temperature and gradually decreasesit to 0, where the program becomes exclusively greedy. Again for simplicity’s sake,we used a linear cooling schedule, but other functions could also be considered.4imension | S | = 488 | S | = 488 Previous boundsExplicit S i ’s Proposition 1.3 (2011)25 197048 197048 19704026 198512 198512 19848027 199976 199968 19991228 204368 204312 20418829 208272 208114 20793030 219984 219368 21900831 232874 231412 230872 Table 4.
New lower bounds for kissing numbers, based on | S | = 488 .The advantage of this algorithm is that it can explore more of the solution spacewithout getting stuck in local extrema. Given a slow enough cooling schedule, thedistribution of possible states converges over time to the global maximum for | S | .This algorithm led to the current record | S | = 488 after starting from an initial set S obtained by calling the greedy algorithm on different orderings of the vectors. Dueto resource constraints, we optimized based on the assumption that S is antipodal(meaning that we added and removed vectors in antipodal pairs ± x ).3. Maximizing Sizes of the Subsets via Explicit Construction
We also explicitly constructed a sequence of mutually disjoint subsets S i withpairwise inner product at most 1 using a probabilistic algorithm analogous to theconstruction of S . Using the algorithm developed in [CLGM + g of the automorphism group Co and repeatedly applied themto S . We did this until gS was disjoint from all the previous sets; if we couldnot find such a gS , we chose the largest set that we could find after deleting theoverlap with previously constructed sets. After several trials and modifications,this approach yielded 51 mutually disjoint subsets of C of nonincreasing size, 24of which have size 488. Using these subsets (which can be found as text files inAppendix A), we obtained the new lower bounds from Theorem 3.1 for the kissingnumbers in 25 through 31 dimensions (the bounds in the second column result fromapplying Proposition 1.3 using | S | = 488 ). Theorem 3.1.
The kissing numbers in 25 to 31 dimensions are bounded below bythe values indicated in Table 4.
Although these computer-generated subsets of C generate the current records,they are almost certainly suboptimal. Furthermore, they display no obvious struc-ture or mathematical explanation; we therefore also present in Section 4 an improve-ment of [CJKT11]’s probabilistic construction of the S j ’s from Proposition 1.3.Even using | S | = 480 , this construction leads to improved kissing number lowerbounds in dimensions 30 and 31 compared to the previous bounds.4. Improved Probabilistic Argument
The following method results in better lower bounds than the original probabilis-tic method used in [CJKT11], although it is surpassed by the explicit constructionsdescribed in Section 3. It assumes that | S | ≤ ; the results when applied to ournew S of size 488 (from Section 2) can be found in Table 5.5ecall that C is the set of 196560 minimal vectors of the Leech lattice Λ . Again,let S = S , where S is an antipodal subset of C such that any two elements x, y ∈ S satisfy h x, y i ≤ .For j ≥ , we let S j be of the form S j = g j S \ [ g j S ∩ ( S ∪ · · · ∪ S j − )] , where g j belongs to the automorphism group Co of the Leech lattice. Since S isantipodal, the S j ’s are also antipodal.We recursively construct lower bounds for | S k +1 | from previously obtained lowerbounds on | S i | , where i ≤ k . Our argument is based on a careful examination ofthe set of automorphism group elements g for which | gS ∩ ( S ∪ · · · ∪ S k ) | is lessthan its average value over the whole group.Using this method, we construct larger S j ’s compared to the ones obtained us-ing the original technique in [CJKT11]. Using | S | = 488 from Section 2, this leadsto improved lower bounds in dimensions 28 to 31, compared to applying Propo-sition 1.3 to the same | S | . It also leads to improved lower bounds in dimensions30 and 31 using | S | = 480 from [CJKT11], compared with the lower bounds giventhere. However, this argument does not yield lower bounds on kissing numbers ashigh as those given by the explicit greedy construction from Section 3.We first provide some definitions: Definition 4.1. (1) Let E k = E g ∈ Co [ | gS ∩ ( S ∪ S ∪ · · · S k ) | ] , where g is chosen uniformly atrandom from Co .(2) Let ⌈ x ⌉ denote the smallest even integer at least x and ⌊ x ⌋ denote thegreatest even integer at most x .(3) For any g ∈ Co , let δ ( g ) = | gS ∩ ( S ∪ · · · ∪ S k ) | − E k .(4) Let G + = { g ∈ Co | δ ( g ) > } and G − = { g ∈ Co | δ ( g ) < } .It follows from X g ∈ G + δ ( g ) + X g ∈ G − δ ( g ) = X g ∈ G ( | gS ∩ ( S ∪ · · · ∪ S k ) | − E k ) = 0 that X g ∈ G + | δ ( g ) | = X g ∈ G − | δ ( g ) | . We can then prove a basic property of δ ( g ) : Lemma 4.2.
One of the following two possibilities holds: (1) max g ∈ G − | δ ( g ) | = E k − ⌊ E k ⌋ , or (2) max g ∈ G − | δ ( g ) | ≥ E k − ⌊ E k ⌋ + 2 .Proof. Due to the antipodality of S , S , . . . , S k , we know g i v ∈ ( S ∪ · · · ∪ S k ) ifand only if g i ( − v ) ∈ ( S ∪ · · · ∪ S k ) . Therefore, | g i S ∩ ( S ∪ · · · ∪ S k ) | is an eveninteger. Also, | δ ( g ) | ≥ E k − ⌊ E k ⌋ for g ∈ G − by definition.In addition, note that G − = ∅ . This is because if G − = ∅ , then G + = ∅ , since P g ∈ G + | δ ( g ) | = P g ∈ G − | δ ( g ) | . However, consider the identity element e ∈ Co . Wehave S = S = eS , so | eS ∩ ( S ∪ · · · ∪ S k ) | = | S | > E k . Therefore, δ ( e ) > and e ∈ G + . So G − = ∅ .It follows that either max g ∈ G − | δ ( g ) | = E k −⌊ E k ⌋ or max g ∈ G − | δ ( g ) | ≥ E k −⌊ E k ⌋ +2 . (cid:3)
6n order to analyze the implications of the two possibilities in Lemma 4.2, wereformulate the problem in terms of graph theory:
Definition 4.3.
Let H be a graph whose vertices correspond to the elements ofCo . Two vertices of H are adjacent if their corresponding elements g i and g j satisfy g i S ∩ g j S = ∅ . Lemma 4.4.
The graph H is regular.Proof. This follows from the fact that Co acts on itself by left multiplication.Consider three elements g , g , and g ′ in Co . Since Co ’s elements are auto-morphisms that act injectively on the Leech lattice, g S ∩ g ′ S = ∅ if and only if g S ∩ g g − g ′ S = ∅ . This is because g g − ( g S ∩ g ′ S ) is empty if and only if ( g S ∩ g ′ S ) is empty. Then composition by g g − preserves edges and thereforerepresents a graph automorphism of H . It immediately follows that the verticescorresponding to the arbitrarily chosen g and g have the same degree, so everyvertex in H has the same degree. (cid:3) Definition 4.5. (1) Let P be the subgraph of H consisting of the vertices corresponding to theelements in G − .(2) Let R be the subgraph of H consisting of the k vertices corresponding tothe elements g , g , . . . , g k associated with the subsets S , . . . , S k .(3) Let Q be the subgraph of H consisting of the rest of the vertices of H . Lemma 4.6.
The subgraphs P and R are vertex disjoint for k ≤ and | S | ≤ .Proof. Recall that each vertex in R corresponds to an element g i ∈ Co , whichis associated with subset S i , where i ∈ { , , . . . , k } . Thus g i S ∩ S i = S i , so | g i S ∩ ( S ∪ · · · ∪ S k ) | ≥ | S i | . Note that the size of each S i is greater than or equalto the size of the corresponding subset generated by applying Proposition 1.3, asmentioned at the beginning of this section. Therefore, for i ≤ k ≤ , we have | S i | ≥ | S | − P i − j =1 | S j ||C| ! > | S | − P kj =1 | S j ||C| ! ≥ | S | (cid:18) − | S ||C| (cid:19) . Since | S | ≤ < |C| = 1965 . , this is greater than | S | | S ||C| > | S | P kj =1 | S j ||C| = E k . Therefore g i is in G + for i = 1 , , . . . , k . But each vertex in P corresponds to anelement in G − . Since G + and G − are disjoint by definition, it follows that P and R are disjoint. (cid:3) Before we carry on analyzing the properties of H , we make use of the followinggeneral graph theory result: 7 emma 4.7. Let Ω be a w-regular graph. Let A and B be disjoint induced subgraphsof Ω such that V ( A ) ∪ V ( B ) = V (Ω) and | V ( A ) | ≥ | V ( B ) | . Then if B contains anedge, A must also contain an edge.Proof. Suppose, for contradiction, that there is at least one edge in B but no edgesin A .Let E ( A, B ) denote the set of edges between A and B . We will calculate thesize of E ( A, B ) in two different ways. First, consider the sum of the degrees of thevertices in A . Since Ω is a w -regular graph, each vertex in A has degree w , so sincethere are no edges in A , | E ( A, B ) | = w | V ( A ) | . Now consider the sum of the degreesof the vertices in B . Each vertex in B also has degree w , but there is at least oneedge in B , so | E ( A, B ) | ≤ ( | V ( B ) | − w + 2( w −
1) = w | V ( B ) | − .Therefore, w | V ( A ) | ≤ w | V ( B ) | − . But | V ( A ) | ≥ | V ( B ) | , so w | V ( A ) | ≤ w | V ( B ) | − ≤ w | V ( A ) | − . Thus we have a contradiction. (cid:3)
We now apply Lemma 4.7 to our situation:
Corollary 4.8.
Suppose that ≤ k ≤ , and | S | ≤ . If | G − | ≥ | G + | , thenthere exist distinct g , g ∈ G − such that g S ∩ g S = ∅ . Proof.
Note that by Definition 4.5, | V ( P ) | = | G − | .Also by Definition 4.5, V ( R ) ∪ V ( P ) ∪ V ( Q ) = V ( H ) . By Lemma 4.6, we knowthat P and R are vertex disjoint, which together with the definition of Q , impliesthat P is vertex disjoint from R ∪ Q .By Proposition 1.3, we know that we can construct | S | ≥ | S | (1 − | S ||C| ) . Thus, bythe antipodality of S , we have | S | = | S | for | S | ≤ . Therefore, since S and S are disjoint and are both of size | S | , the vertices corresponding to g and g areadjacent. Since g and g are both in R (as k ≥ ), they are also in R ∪ Q . Thusthere is an edge in R ∪ Q .Additionally, as proved in Lemma 4.4, H is a regular graph.Then since | G − | ≥ | G + | , we have | V ( P ) | ≥ | V ( R ∪ Q ) | . Thus by Lemma 4.7,there must be an edge in P .We let the vertices connected by this edge be g and g , completing the proof. (cid:3) We remark that Lemma 4.6 and therefore Corollary 4.8 hold for values of k muchlarger than 50; we state the condition k ≤ because, from Table 3, we only need k ≤ to construct the kissing number lower bounds in dimensions 25 to 31.We now apply Corollary 4.8 to describe the implications of Lemma 4.2 part (1) inthe next two lemmas: Lemma 4.9.
Suppose max g ∈ G − | δ ( g ) | = E k − ⌊ E k ⌋ . Then | G − | ≥ | Co | ⌈ E k ⌉ − E k ) . Proof.
Since max g ∈ G − | δ ( g ) | = E k − ⌊ E k ⌋ , we have | δ ( g ) | = E k − ⌊ E k ⌋ g ∈ G − . We also have X g ∈ G + | δ ( g ) | = X g ∈ G − | δ ( g ) | = | G − | ( E k − ⌊ E k ⌋ ) . Now, by definition, | δ ( g ) | ≥ ⌈ E k ⌉ − E k for g ∈ G + . Also, since max g ∈ G − | δ ( g ) | = E k − ⌊ E k ⌋ , by the definition of G − , E k = ⌊ E k ⌋ . Thus E k is not an even integer.However, | gS ∩ ( S ∪ · · · ∪ S k ) | is an even integer by antipodality, so δ ( g ) = 0 forall g ∈ Co . Therefore, G + ∪ G − = Co . Therefore, | G − | ( E k − ⌊ E k ⌋ ) = X g ∈ G + | δ ( g ) |≥ ( | Co | − | G − | )( ⌈ E k ⌉ − E k ) . This implies that | G − | ( E k − ⌊ E k ⌋ + ⌈ E k ⌉ − E k ) ≥ | Co | ( ⌈ E k ⌉ − E k ) , which simplifies to | G − | ≥ | Co | ⌈ E k ⌉ − E k ) . (cid:3) Lemma 4.10.
Let ≤ k ≤ and | S | ≤ . Suppose that max g ∈ G − | δ ( g ) | = E k −⌊ E k ⌋ and | G − | ≥ | G + | . Then there exist S k +1 , S k +2 ⊂ C such that | S k +1 | , | S k +2 | ≥ | S | − ⌊ E k ⌋ , and S ℓ ∩ S m = ∅ for all distinct ℓ and m satisfying ≤ ℓ, m ≤ k + 2 .Proof. We have | G − | subsets, gS , such that E k − | gS ∩ ( S ∪ · · · ∪ S k ) | = E k − ⌊ E k ⌋ , in other words | gS \ ( gS ∩ ( S ∪ · · · ∪ S k )) | = | S | − ⌊ E k ⌋ . Now, since | G − | ≥ | G + | , at least two of the | G − | subsets are disjoint by Corol-lary 4.8. Call them g α S and g β S . Now set S k +1 = g α S \ ( g α S ∩ ( S ∪ · · · ∪ S k )) and S k +2 = g β S \ ( g β S ∩ ( S ∪ · · · ∪ S k )) . (cid:3) We consider the implications of Lemma 4.2 part (2):
Lemma 4.11.
Suppose that max g ∈ G − | δ ( g ) | ≥ E k − ⌊ E k ⌋ + 2 . Then there exists asubset S k +1 ⊂ C such that | S k +1 | ≥ | S | + 2 − ⌊ E k ⌋ and S ℓ ∩ S m = ∅ for all distinct ℓ and m satisfying ≤ ℓ, m ≤ k + 1 . roof. Suppose that max g ∈ G − | δ ( g ) | ≥ E k − ⌊ E k ⌋ + 2 . Then there exists an element g in G − such that E k − | gS ∩ ( S ∪ · · · ∪ S k ) | ≥ E k − ⌊ E k ⌋ + 2 | gS ∩ ( S ∪ · · · ∪ S k ) | ≤ ⌊ E k ⌋ − | gS \ ( gS ∩ ( S ∪ · · · ∪ S k )) | ≥ | S | + 2 − ⌊ E k ⌋ . Now set S k +1 = gS ∩ ( S ∪ · · · ∪ S k ) . (cid:3) Having analyzed both possibilities presented in Lemma 4.2, we now derive inLemma 4.12 and Corollary 4.13, a set of consequences on | S k +1 | , | S k +2 | and E k +2 under the assumption that ⌊ E k ⌋ is even. These results will be crucial in constructingan algorithm to derive our new kissing number lower bounds. Lemma 4.12.
Let ≤ k ≤ and | S | ≤ . If ⌊ E k ⌋ is even, then either (1) There exist S k +1 , S k +2 ⊂ C such that | S k +1 | , | S k +2 | ≥ | S | − ⌊ E k ⌋ or (2) There exist S k +1 , S k +2 ⊂ C such that | S k +1 | ≥ | S | − ⌊ E k ⌋ + 2 and | S k +2 | ≥| S | − ⌊ E k ⌋ − .Proof. Let ⌊ E k ⌋ be even.By Lemma 4.2 either(1) max g ∈ G − | δ ( g ) | = E k − ⌊ E k ⌋ , or(2) max g ∈ G − | δ ( g ) | ≥ E k − ⌊ E k ⌋ + 2 .If (1) holds, then since ⌊ E k ⌋ is even, ⌈ E k ⌉ − E k ≥ , so by Lemma 4.9, | G − | ≥ | Co | . Thus by Lemma 4.10, there exist subsets S k +1 and S k +2 of C such that | S k +1 | , | S k +2 | ≥ | S | − ⌊ E k ⌋ .If (2) holds, then by Lemma 4.11, there exists a subset S k +1 of C such that | S k +1 | ≥ | S | − ⌊ E k ⌋ + 2 . Then, by Proposition 1.3, we can find an S k +2 such that | S k +2 | ≥ | S | − P k +1 j =1 | S j ||C| ! = | S | − P kj =1 | S j ||C| − | S k +1 ||C| ! = | S | − E k − | S | | S k +1 ||C| . Since | S k +1 | ≤ | S | , for | S | ≤ we have | S | | S k +1 ||C| ≤ | S | |C| < 2. Therefore, | S k +2 | ≥| S | − E k − , and since | S k +2 | is an even integer by the antipodality of the | S | , wehave | S k +2 | ≥ | S | − ⌊ E k ⌋ − , thus proving the lemma. (cid:3) We remark that although we do not know whether case (1) or (2) of Lemma 4.2occurs (and which of cases (1) and (2) in Lemma 4.12), we show in the next corol-lary that the two cases imply the same consequences in terms of E k +2 , and suitableinequalities in linear combinations of | S j | ’s pertaining to Table 3 for generating kiss-ing number lower bounds. This yields Algorithm 4.14 for the recursive constructionof kissing number lower bounds. 10 orollary 4.13. Let | S | ≤ and ≤ k ≤ . If ⌊ E k ⌋ is even, then there exist S k +1 , S k +2 ⊂ C such that(4.1) E k +2 = | S | P kj =1 | S j | + 2( | S | − ⌊ E k ⌋ ) |C| , (4.2) | S k +1 | + | S k +2 | ≥ | S | − ⌊ E k ⌋ ) and(4.3) | S k +1 | + | S k +2 | ≥ | S | − ⌊ E k ⌋ ) . Proof.
By Lemma 4.12, if ⌊ E k ⌋ is even, then there exist S k +1 , S k +2 ∈ C such thateither(1) | S k +1 | , | S k +2 | ≥ | S | − ⌊ E k ⌋ , or(2) | S k +1 | ≥ | S | − ⌊ E k ⌋ + 2 and | S k +2 | ≥ | S | − ⌊ E k ⌋ − .If case (1) holds, then we can delete elements from S k +1 and S k +2 until we have | S k +1 | = | S k +2 | = | S | − ⌊ E k ⌋ .If case (2) holds, then we can delete elements from S k +1 and S k +2 until we have | S k +1 | = | S | − ⌊ E k ⌋ + 2 and | S k +2 | = | S | − ⌊ E k ⌋ − .Then, clearly the first and third conditions are satisfied in both cases. Thesecond condition is also satisfied because in both cases, we have | S k +1 | + | S k +2 | =2( | S | − ⌊ E k ⌋ ) , so E k +2 = | S | P k +1 j =1 | S j ||C| = | S | P kj =1 | S j | + 2( | S | − ⌊ E k ⌋ ) |C| . (cid:3) The preceding discussion allows us to use Lemma 4.12 to improve on the prob-abilistic construction (Lemmas 7.4 and 7.5) in [CJKT11] as follows:
Algorithm 4.14. (1) Set S = S and bound | S | below using Proposition 1.3.(2) Assume we have constructed S , S , . . . , S k .(3) If ⌊ E k ⌋ is even, Corollary 4.13 ensures the existence of S k +1 and S k +2 suchthat | S k +1 | and | S k +2 | satisfy the bounds given by (4.2) and (4.3). Then E k +2 is also given by Corollary 4.13 to be as in (4.1). Repeat from step(2), using the bounds on S , . . . , S k +2 .(4) If ⌊ E k ⌋ is odd, then bound | S k +1 | below using Proposition 1.3 and continuefrom step (2), using the bounds on S , . . . , S k +1 .(5) Stop when k > .We remark that Algorithm 4.14 is an algorithm for the recursive determinationof kissing number lower bounds but it does not give specific lower bounds for theindividual | S j | ’s, since there are two possible cases in Lemma 4.12 for lower boundson | S k +1 | and | S k +2 | .However, Corollary 4.13 and the sums shown in Table 3 for the kissing numberbounds imply that assuming the lower bounds for both | S k +1 | and | S k +2 | to be | S |−⌊ E k ⌋ (i.e., case (1) in Lemma 4.12) gives at most as good a kissing lower bound11imension Kissing number, | S | = 480 Kissing number, | S | = 488
28 204188 20431629 207930 20812030 219012 21938031 230880 231428
Table 5.
Lower bounds for kissing numbers produced by Algo-rithms 4.14 and 4.15and the same E k +2 as in case (2). Since Algorithm 4.14 after Step (3) depends onlyon E k +2 , we can, for the purpose of generating valid kissing lower bounds, assumethat case (1) occurs. Therefore, we introduce Algorithm 4.15, assuming in Step(3) of Algorithm 4.14 that case (1) always occurs. Algorithm 4.15 will generatethe same valid kissing lower bounds as Algorithm 4.14. These lower bounds mustbe considered only as computational devices for computing kissing lower bounds,and S j ’s of these sizes do not necessarily exist. Rather, Corollary 4.13 gives alower bound for the sums of linear combinations of adjacent disjoint subsets in thesequence, and these bounds are used to give a final lower bound from the sumsgiven in Table 3. The bounds computed by this new algorithm give a final lowerbound which is certainly at most the lower bound given by the legitimate lowerbounds on the | S i | ’s, which is therefore correct, despite the fact that existence ofsubsets of sizes given by this algorithm is not guaranteed. Algorithm 4.15. (1) Set S = S and bound | S | below using Proposition 1.3.(2) Assume that we have lower bounds for | S | , | S | , . . . , | S k | .(3) If ⌊ E k ⌋ is even, case (1) of Lemma 4.12 gives | S k +1 | , | S k +2 | ≥ | S | − ⌊ E k ⌋ ,and Corollary 4.13 gives E k +2 . Repeat from step (2), using the bounds on S , . . . , S k +2 .(4) If ⌊ E k ⌋ is odd, then bound | S k +1 | below using Proposition 1.3 and continuefrom step (2), using the bounds on S , . . . , S k +1 .(5) Stop when k > . Theorem 4.16.
Algorithm 4.15 produces the lower bounds for kissing numbers indimensions 28 through 31 shown in Table 5, using | S | = 480 and | S | = 488 . Al-though the bounds given by Algorithm 4.15 for the | S i | ’s are not necessarily correct,the true bounds given by Algorithm 4.14 give final kissing number bounds at leastas high as these, which means the results in Table 5 are true lower bounds for thekissing numbers in R n for ≤ n ≤ .Proof. The lower bounds for the | S i | ’s produced by Algorithm 4.15 using both | S | = 488 and | S | = 480 are shown in Table 6. We have also included the subsetsizes produced by applying Proposition 1.3, for comparison.Using Table 3 in Section 1.3, these subsets give the kissing number lower boundsshown in Table 7. Since Algorithm 4.15 generates the same kissing number lowerbounds as Algorithm 4.14, this proves the theorem. (cid:3) We remark that by the nature of the argument in Lemmas 7.4 and 7.5 of[CJKT11], the size of the S i ’s decays as i increases, with the rate of decay be-ing larger for larger values of | S | . Therefore, Algorithm 4.14 and Algorithm 4.1512 Lower bound for | S k | [CJKT11], | S | = 480 Alg. 4.15, | S | = 480 [CJKT11], | S | = 488 Alg. 4.15, | S | = 488 Table 6.
Lower bounds on the sizes of the hypothetical S i ’s gen-erated by Algorithm 4.15 13imension [CJKT11] | S | = 480 Alg. 4.15, | S | = 480 [CJKT11], | S | = 488 Alg. 4.15, | S | = 488
28 204188 204188 204312 20431629 207930 207930 208114 20812030 219008 219012 219368 21938031 230872 230880 231412 231428
Table 7.
Kissing number lower boundsyield larger improvements for larger values of | S | . This is demonstrated by theeffect of Algorithm 4.14 on the kissing number lower bounds for | S | = 480 and | S | = 488 shown in Table 7. We thus note that Algorithm 4.15 will yield evenlarger improvements if we find a larger S in the future.5. Future Directions
The Structure of the Leech Lattice and its Automorphisms.
We sus-pect that an optimal S should exhibit symmetries not currently in the S of size 488.Also worth noting is the surprising fact that we could achieve 24 disjoint subsets ofsize 488—this suggests a deeper structure and raises the possibility that we can domuch better than the greedy algorithm, with a th or even st disjoint copy. Evenchanging the greedy algorithm to a backtracking one might increase the number ofdisjoint subsets we can fit, although the running time might become an issue.5.2. Graph-theoretic approaches.
We might translate the problem to a max-imum independent set problem or a maximum clique problem and approach theproblem graph-theoretically. These problems are generally NP-complete, and wewould still need clever techniques adapted to symmetric graphs, approximationalgorithms, or other graph-theoretic heuristics to make this approach fruitful.5.3.
Improving current computational methods.
The algorithms we usedcould be optimized in a number of ways. New search heuristics may result in alarger S , as may a simply longer simulated annealing with a slower temperaturedecrease using more computational power and time. The explicit construction of S i can almost certainly be improved. Given that we used a greedy algorithm, westrongly doubt that the S i ’s are anywhere near optimal; a clever modification tohow the program traverses the search tree will most likely improve the result. Appendices A. Data Files
The data files associated with the explicit constructions described in this papercan be found in the Git repository at https://github.com/kenzkallal/Kissing-Numbers .The files of the form
S_i.txt contain the lists of vectors in the subsets S i . We alsoinclude the file minvects.txt , containing our ordering of the vectors of C , as wellas Vbasis.txt , which is the basis we used for Λ . Acknowledgements
The results in this paper originated from a research lab project in PROMYS2015. We are deeply grateful to Henry Cohn for proposing this problem, and for14is mentoring. We also thank our PROMYS research lab counselor Claudia Fengfor her support, as well as Erick Knight, David Fried, Glenn Stevens, the PROMYSFoundation, and the Clay Mathematics Institute for making this research possible.
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