Complexity and algorithms for computing Voronoi cells of lattices

COMPLEX ITY AND A L GORITHMS FO R COMP U TING V OR ONOI CELLS OF LA TTICES MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALLENTIN A B S T R A C T . In this paper we are concerned with finding the vertices of the V oronoi cell of a Euclidean l attice. Giv en a basis of a lattice, we prov e that computing the number of vertices is a # P -hard problem. On the other hand we describe an algorithm for this pro blem which is especially suit ed for lo w dimen- sional (say dimensions at most 12 ) and for highly-symmetric lattices. W e use our implemen tation, which drastically outperforms those of curren t computer al- gebra systems, to fi nd the vertices of V oronoi cells and quantizer constants of some prominent lattices. 1. I N T R O D U C T I O N Let L = B Z m ⊆ R n be a lattice of rank m in Euclidean space giv en by a matrix B ∈ R n × m of rank m . By lin L we denote the linear subspace spanned by the elemen ts of L . The V or onoi c ell of L is V( L ) = { x ∈ lin L : k x k ≤ k x − v k for all v ∈ L } . The V oronoi cell of a lattice is a centrally symmetric , con ve x polytope. The poly- topes V ( L ) + v for v ∈ L tile lin L . The study of V o ronoi cells i s moti vated b y the fact that m ost i mportant g eometric l attice pa rameters ha ve a direct interpretati on in terms of the V oronoi cell: The determinan t det L equals the volume of V ( L ) , the pac king radius λ ( L ) equal s the inradius of V( L ) , the covering r adius µ ( L ) equ als the circ umradius of V( L ) , and the quantizer cons tant G ( L ) is G ( L ) = (det L ) − (1+2 /n ) Z V( L ) k x k 2 dx. In this p aper we cons ider theoret ical and practical aspect s of the computat ion of the cov ering radius as w ell as the quantizer constant of a lattice. These two pa- rameters ha ve many appli cation s, we just name a fe w: By computin g the coveri ng Date : Nov ember 26, 2024. 1991 Mathematics Subject Classification. 03D15, 11H56, 11H06, 11B1, 52B55, 52B12. K e y wor ds and phrases. lattice, V oronoi cell, Delone cell, cov ering radius, quantizer constant, lattice isomorphism problem, zonotop e. The work of the first author has been supported by the C roatian Ministry of Science, E ducation and Sport under contract 098 -0982705 -2707. The second and the third author were sup ported by the Deutsche Forschungsgemeinsch aft (DFG) und er grant S CHU 1503/4-2. The third author was also supported by the Netherlands Orga nization for S cientific Research under grant NWO 639.032.203. All three authors thank the Hau sdorf f Research Institute for Mathematics for its hospitality and support. 1 2 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN radius , we get an uppe r bound fo r the lattice sphere c ov ering p roblem, which i s t he proble m of minimizi ng the co v ering radiu s among the l attices of fi xed determinan t (see [CS99, Cha pter 2] an d [SV 06]). T he computatio n of the co v ering radius of the Leech latti ce in [CS99, Chapter 23] had a major impact on the stud y of hyperbolic reflection group s (see [CS99, Chapter 27]). An upper bound for the Frobeniu s number of a set of integer s can be obtained from the cov ering radius of a suitable lattice (see [FR05]). A recen t applic ation comes from public ke y cryptograp hy; Miccianc io [Mic0 4] foun d a ne w c onnect ion between th e av erage-c ase complex - ity of finding the packi ng radius and the worst-ca se complex ity of determin ing the cov erin g radius. In information theory , the quality of a latti ce as a v ector quan tizer is determined by its quan tizer constan t (see [G G92, ELZ05, SB03] and [CS99, Chapter 2.3, Chapte r 21]). The structure of this paper is as follo ws. In Section 2 we discus s the com- putati onal comple xity of the coveri ng radius pr oblem. W e prove th at the related proble m of counti ng ver tices o f the V o ronoi cell is # P -hard. As a byprod uct of our construction , we sho w that the lattic e is omorphis m problem is at least as dif- ficult as the graph isomorphism pro blem. W e turn to practical al gorithms for th e cov erin g radius proble m in Section 3. There we describe an algor ithm which com- putes the vertices of the V orono i cell of a lattice. Based on this algorith m we giv e an algorithm for comput ing the quantizer constant in Section 4. In Section 5 we report o n computations with our implementatio n. W e d etermine the exact cover ing radius and quantiz er c onstan ts of many prominent lattices w hich w ere not kno wn before . 2. C O M P U T A T I O N A L C O M P L E X I T Y W e formulate the cove ring radius problem as a decision problem. Pro blem 1. Covering radius pr oblem Input: m , n , B ∈ Q m × n , µ ∈ Q . Output: Y es , if µ ( B Z n ) ≤ µ , No otherwise . It is conjectu red (see [Mic04, Section 1.1]) that the cover ing radius problem is NP -hard. Hav i v and Rege v [HR06] sho wed that there is a constant c p so that the cov erin g radius in the l p -norm is Π 2 -hard to approximat e w ithin a constan t less than c p for any lar ge enough p . In [GMR05] Guruswami, Micc iancio and R ege v pro ved that app roximatin g it within a facto r of O ( p m/ log m ) for a lattice of rank m cannot be NP -hard unless the polynomial hierarchy collapses . Currently , there is only one kno wn general and practi cal method to comput e µ ( L ) for a lattice L : First one enumerates the vertices of V ( L ) an d then one fi nds the verte x with large st norm. The n umber of v ertic es o f V ( L ) can be as large as ( m + 1)! and f urthermo re, as we sho w in Theore m 1, ev en co mputing this number is # P -hard. Pro blem 2. V ertices of a lattice V or onoi cell Input: m , n , B ∈ Q m × n . Output: Number of vertices of V ( B Z n ) . COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 3 Theor em 1. The pr oblem “V ertices of a lattice V or onoi cell” is # P -har d. It will be obvi ous from the proo f that we could restrict the probl em to the case m = n . W e reduce the problem “ Acycli c orien tations of a graph ”, which Linial [Lin86] sho w ed to be # P -complete, to Problem 2. Pro blem 3. Acyclic ori entatio ns of a graph Input: A gra ph G = ( V , E ) . Output: The number of ori entatio ns of G with no dir ected cir cuit. The structure of the proof of T heorem 1 is as follows: In Section 2.1 we con- struct a matrix B w ith columns index ed by E defining a lattice L ( G ) = B Z E from G in p olynomia l time. T hen we s ho w that the v ertice s of the V oron oi cell of V ( L ( G )) are in bijection with the acyclic orientation s of G . T o est ablish th is bijecti on we ne ed se veral int ermediate steps . I n Secti on 2.2 we as sociate to G a hyperp lane arrange ment H ( G ) whos e ch ambers are in biject ion with the acyclic orient ations of G . In Section 2.3 we recall th at the chambers of a hyperplane ar- rangemen t are in bijection with the vertices of a zonotope associat ed to the hyper - plane arran gement. Thes e two steps are stand ard and we co ver them rathe r briefly . In Section 2.4 w e sho w that the V orono i cell of L ( G ) is a zonotope which, up to a linear transfo rmation, is the one associated to the hype rplane arrang ement H ( G ) . In Sectio n 2.5, as a byprodu ct of this constructi on, we sho w that the lattice iso- morphism pr oblem is at least a s dif ficult as the gr aph isomor phism proble m. Some related comple xity results concerning ver tex enumerat ion of polyhe dra gi ve n by linear ineq ualitie s are in [KBBEG08, Dy83]. 2.1. F r om graphs to lattices. Let G = ( V , E ) be a conne cted graph with verte x set V = { 1 , . . . , n } and edge set E . W e consider the follo w ing orien tation of the edges of G : The he ad of an edge e = { v , w } ∈ E is e + = max { v , w } and the tail is e − = min { v , w } . Let T ⊆ E be th e ed ge set of a spanning tree of G , and let e ∈ T . Deleting e from T di vid es T into two conn ected components w ith verte x sets T + e and T − e , where e + ∈ T + e and e − ∈ T − e . Define the vec tor b T ,e ∈ Z E by b T ,e ( f ) =    1 , if f + ∈ T + e and f − ∈ T − e , − 1 , if f − ∈ T + e and f + ∈ T − e , 0 , otherwis e. Then L ( G, T ) = ( X e ∈ T α e b T ,e : α e ∈ Z ) ⊆ Z E is a lattic e of rank n − 1 . Pro position 1. Let T and T ′ be spanning tr ees of G . Then , L ( G, T ) = L ( G, T ′ ) . Pr oof. Since one can con nect any two spanni ng trees by a sequ ence of tran sfor - mations of the form T ↔ T \ { e } ∪ { f } it suf fices to pro ve the propositio n for T ′ = T \{ e } ∪ { f } . Let g ∈ T ′ . If g = f , then b T ′ ,f = ± b T ,e . If g ∈ T , then denote 4 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN by C the cycl e containing e and f . If g / ∈ C then b T ′ ,g = b T ,g . T he subgrap h of G with edge set T \ { e, g } has three connec ted components, denoted by C 1 , C 2 , C 3 . Giv en h = { v , w } ∈ E , the value of b T ′ ,g ( h ) , b T ,g ( h ) and b T ,e ( h ) depends only on whic h conne cted component v and w belon g to. So, in compu ting b T ′ ,g , we can reduce oursel ves to the cas e whe n G is the co mplete graph o n { 1 , 2 , 3 } , g = { 1 , 3 } , e = { 1 , 2 } an d f = { 2 , 3 } . Easy computa tion giv es b T ′ ,g = b T ,g + b T ,e and so we conclu de that b T ′ ,g = b T ,g + αb T ,e with α ∈ {− 1 , 0 , +1 } .  In the fol lo wing we omit t he spanning tree T from the notatio n L ( G, T ) and just write L ( G ) . Note that one can find a basis of L ( G ) gi ven G in polynomial time. 2.2. F r om graphs to hy perplane arrangements. A matrix (1) V = ( v 1 , . . . , v m ) ∈ R n × m with non-z ero column ve ctors v i ∈ R n gi ve s an arran gemen t of hype rplane s H ( V ) = { H 1 , . . . , H m } with H i = { c ∈ R n : c · v i = 0 } . The hyperp lane arrang ement H ( V ) div ides the space R n into polyhe dral cones, called r e gions , of diffe rent dimensions . The region s are partiall y ordered by inclu- sion and full-d imension al regi ons are called chambe rs . T o associat e a hyper plane arrangemen t H ( G ) with G we consid er the incide nce matrix D G ∈ R V × E of G w hich is gi v en by D G ( v , e ) =    1 , if v = e + , − 1 , if v = e − , 0 , otherwis e. Then we define the hyperp lane arrangement of G by H ( G ) = H ( D G ) . In [GZ83, L emma 7.1] Greene and Zasla vsk y sho w that the chambers of H ( G ) are in bijection w ith the acyclic orientatio ns of G : Let ~ E be an acycl ic orientati on of E . Then a chamber of H ( G ) is gi ven by Reg( ~ E ) = { x ∈ R V : x v < x w if ( v , w ) ∈ ~ E } . Let R be a chamber of H ( G ) . Then a n acycli c orientation of E is g i ven by ~ E ( R ) = { ( v , w ) : { v , w } ∈ E and x v < x w for e very x ∈ R } . Obvio usly , Reg ( ~ E ( R )) = R . 2.3. H yperplane arrange ments and zonotopes. The matrix V in (1) defines a zonoto pe Z ( V ) by Z ( V ) = ( m X i =1 α i v i : − 1 ≤ α i ≤ 1 ) . The faces of Z ( V ) are partially ordered by inclusion. It is a well-kn o wn fac t (see e.g. [Zie95, Theorem 7.16]) that the partially ordere d set of regio ns of the hyper - plane arrang ement H ( V ) is anti-isomorph ic to the partiall y ordered set of face s COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 5 of Z ( V ) : Let R be a reg ion of H ( V ) . Let x ∈ R . T hen the correspo nding fac e F ace( R ) of Z ( V ) gi ven by F ace( R ) =  y ∈ Z ( V ) : x · y = max z ∈ Z ( V ) x · z  , does not depen d on the cho ice of x . L et F be a face of Z ( V ) . Let y be in the relati ve interior of F . Then the corresp ondin g regio n Reg ( F ) of H ( V ) gi ven by Reg( F ) =  x ∈ R n : max z ∈ Z ( V ) x · z = x · y  , does not depend on the choice of y . Obviously , F ace(Reg ( F )) = F and F ′ ⊆ F if and only if Reg ( F ′ ) ⊇ Reg ( F ) . In particular , the chambers of H ( V ) are in bijecti on with the vert ices of Z ( V ) . 2.4. F r om lattices to zonotopes. Let L ⊆ R n be a lattice. The s uppor t of a vector v ∈ L is v = { i ∈ { 1 , . . . , n } : v i 6 = 0 } . The v ecto r v is cal led elementa ry if v ∈ {− 1 , 0 , +1 } n \ { 0 } and if v has minimal support among all vec tors in L \ { 0 } . W e say that two vecto rs v , w ∈ L are confor mal if v i w i ≥ 0 for all i = 1 , . . . , n . The lattice L is called r e gular if for eve ry vector v ∈ L \ { 0 } there exist s an elementa ry vector u ∈ L with u ⊆ v . Lemma 1. ([T ut71, Chapter 1]) (i) F or any gra ph G the lat tice L ( G ) is r e gular . (ii) If L is a r e gular lattice , then every v ∈ L can be written as a sum of pairwise confor mal elementa ry vector s. (iii) If L is a r e gular lattice, v ∈ L is el ementary , and u ∈ L satisfies u = v , then ther e exists a facto r α ∈ Z suc h that u = αv . A vecto r v ∈ L for which V( L ) ∩ { x ∈ R n : x · v = 1 2 v · v } is a facet of V( L ) is called rel evan t . V oronoi characte rizes in [V or08, page 277] the relev ant vect ors of L : A nonzero v ecto r v ∈ L is relev ant if and only if ± v are the only sho rtest vec tors in v + 2 L . Pro position 2. In a r e gular lattice , a vector is elementa ry if and only if it is re le- vant. Pr oof. Let v ∈ L be a relev ant vect or . By Lemma 1 (ii), we can write v = P m k =1 w k as a sum of pairwise c onformal elementary vecto rs w k ∈ L . Assume that m ≥ 2 . Defining u = v − 2 w 1 gi ve s u 6 = ± v and u · u = v · v − 4( v − w 1 ) · w 1 . Since the v ectors w k , k = 1 , . . . , m , are pairwise conformal we ha ve ( v − w 1 ) · w 1 ≥ 0 , and ± v is not the unique shortest vecto r in v + 2 L . In this case v cannot be a rele v ant vect or . H ence, m = 1 an d v is an element ary vect or . Let v ∈ L be an elementary vector , and let u ∈ v + 2 L be a lattice vector with u 6 = ± v . W e hav e v − u ∈ 2 L ⊆ 2 Z n and v i ∈ {− 1 , 0 , +1 } , which sho w s v ⊆ u . The case v 6 = u immediately l eads to v · v < u · u . If v = u , then by Lemma 1 (iii), there e xists a f acto r α ∈ Z \ {− 1 , +1 } so that u = αv , hence v · v < u · u . In both cases ± v are the onl y shorte st vecto rs in v + 2 L . Hen ce, v is a rele v ant v ector .  6 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN The follo wing special case of the Farkas le mma is prov ed e.g. in [Roc7 0, Theo- rem 22.6]. Lemma 2. Let L ⊆ R n be a r e gular lattice . Let x ∈ R n be a vecto r , and let α 1 , . . . , α n ∈ R ∪ {±∞} . Eith er ther e exi sts a vecto r y ′ ∈ (lin L ) ⊥ lying in x + Q n i =1 [ − α i , α i ] , or ther e exists a vecto r y ∈ lin L suc h th at for all z ∈ x + Q n i =1 [ − α i , α i ] the inequ ality y · z > 0 holds. If the second conditio n holds, then one can ch oose y to be an elementary vecto r of L . Theor em 2. Let L ⊆ R n be a r e gular lattice . Let P ∈ R n × n be the matrix of the ortho gona l pr oject ion of R n onto lin L . Then, V( L ) = 1 2 Z ( P ) = P ([ − 1 / 2 , 1 / 2] n ) . Pr oof. Suppose that x ∈ [ − 1 / 2 , 1 / 2] n . For all v ∈ Z n \ { 0 } the inequali ty x · v ≤ 1 2 v · v ho lds. W rite x = y + y ′ with y = P x ∈ lin L and y ′ ∈ (lin L ) ⊥ . For all v ∈ L \ { 0 } we ha ve y · v = x · v − y ′ · v ≤ 1 2 v · v . Thus , P x ∈ V( L ) . Suppose no w that y ∈ V ( L ) . If there exist s x ∈ ( − y + [ − 1 / 2 , 1 / 2] n ) ∩ (lin L ) ⊥ , then y + x ∈ [ − 1 / 2 , 1 / 2] n and P ( y + x ) = y . Assume that su ch a v ecto r do es not exist. Then by Lemma 2 there is an elementary la ttice ve ctor v ∈ L so that v · ( − y + [ − 1 / 2 , 1 / 2] n ) > 0 . T his implies v · ( − y − 1 2 v ) > 0 . Henc e, − y 6∈ V( L ) . Since V( L ) is centra lly symmetric, this con tradicts the assumpti on y ∈ V( L ) .  In [Big97, Proposition 8.1] Biggs shows that for the lattice L ( G ) the matrix P can be written in the form P = X D G where D G ∈ R V × E is the incidence matrix of G and X ∈ R E × V is gi ven by (2) X ( e, v ) = number of spanni ng trees T with e ∈ T and v ∈ T + e number of spanni ng trees of G . Furthermor e, th e linear map gi ven by X restrict ed to the image of D G is a bijection . Thus, the zonot ope Z ( P ) which is the V orono i cell of L ( G ) eq uals 1 2 X Z ( D G ) . Hence, there is a linear isomorphism between the faces of V( L ( G )) and those of Z ( D G ) . This completes the proof of Theorem 1. Using a straigh tforwa rd computation we get the follo wing propositi on. Pro position 3. Using the notation in (2) , the cover ing radius of the lattice L ( G ) is given by (3) µ ( L ( G )) 2 = max x ∈ [ − 1 / 2 , 1 / 2] E X e ∈ E   X f ∈ E ( X ( e, f + ) − X ( e, f − )) x ( f )   2 . Unfortun ately , we do no t h a ve a co mbinator ial interpre tation of (3). Finding o ne could lead to a proo f of the NP -hardn ess of the cov erin g radius problem. 2.5. L attice isomorphism pro blem. Using the constructi on L ( G ) used in the proof of T heorem 1, we reduce the graph isomorphism problem to the lattice iso- morphism pro blem in po lynomial time. W e don’ t know whet her one can giv e a re ve rse polyn omial time reductio n. For the g raph isomorph ism pr oblem no poly- nomial time a lgorith m is kno w n. It is gen erally belie v ed to lie in NP ∩ co- NP . COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 7 So it is unlik ely that it is NP -hard. For m ore infor mation on the compu tationa l comple xity of this proble m, see the book [KST93] of K ¨ obler , Sch ¨ oning and T ´ oran. Pro blem 4. Lattice iso morphism pr oblem Input: m , n, B , B ′ ∈ Q m × n matrices of rank m . Output: Y es , if ther e is an orthogon al transf ormation O so that O B Z n = B ′ Z n , No oth erwise . Pro blem 5. Graph isomorphis m pr oble m Input: G rap hs G = ( V , E G ) , H = ( V , E H ) . Output: Y es , if ther e is a permutation σ : V → V so that for all v , w ∈ V we have { v, w } ∈ E G if and only if { σ ( v ) , σ ( w ) } ∈ E H , No oth erwise . Theor em 3. Ther e is a polynomial time red uction of the gr aph isomorph ism pr ob- lem to the lattice iso morphism pr oblem. Pr oof. Let G = ( V , E G ) and H = ( V , E H ) be graphs. W e modify G and H by adding thr ee extra v ertices to V each adjace nt to all vertices in V . W e call the ne w graphs G ′ and H ′ which are by constructio n 3 -connect ed and they are isomorphic if and only if G and H are isomorph ic. It is c lear that the lattic es L ( G ′ ) and L ( H ′ ) defined in S ubsect ion 2.1 are iso- morphic whene v er G ′ and H ′ are. For this dire ction it wou ld be e nough to consider the orig inal graphs G and H . No w suppos e that the lattices L ( G ′ ) and L ( H ′ ) are isomorp hic. W e apply the 2-Isomor phism-Theo rem of Whitney (actually w e only use the easy subca se of 3 -conn ected graphs [Oxl92, Lemma 5.3.2]): Because the graphs G ′ and H ′ are 3 -conn ected and there is a bijecti on between the elementary vectors preserving confor mality , the graphs G ′ and H ′ are isomorph ic.  3. A L G O R I T H M S In this section we describ e an algorith m which computes all vertice s of a lattice V oron oi cel l. Our focus is on imp lementabi lity and pra ctical perf ormance, using the symmetries of the lattice. In fact, the algorithm comput es all full-d imension al Delone c ells an d th e adjac encies between them up to equi valenc e. W e giv e ne c- essary definition s in Section 3.1. In Section 3.2 w e describe the algorithm’ s main steps and in the follo wing sections we giv e details about its subalg orithms. In Section 3.6, we explain how to use Gram mat rices in stead of lattice b asis and in Section 3.7 we compar e our method with exis ting algorithms. 3.1. N otatio n. From no w on, we assume lattice s L ⊆ R n to ha v e full rank n . T o encode the vertices of V ( L ) we use D elone cells. A point x ∈ R n defines a Delone cell D ( x ) by D ( x ) = conv  v ∈ L : k x − v k = min w ∈ L k x − w k  . Denote by S ( x, r ) the sph ere with center x and rad ius r . For r = min v ∈ L k x − v k , the s phere S ( x, r ) is called empty , s ince ther e is no lattice p oint ins ide. In this ca se 8 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN the polytope D ( x ) is the con vex hull of S ( x, r ) ∩ L . The Delone cell of a vertex of V( L ) is character ized among all D elone cells by the follo wing propertie s: The origin is a ve rtex of D ( x ) and D ( x ) is ful l-dimens ional. It is well known (see e.g. [Ede01]) that the Delone cells are the project ions of the f aces of the infinite ( n + 1) -dimensiona l polyhedral set Lift( L ) = con v  ( x, k x k 2 ) : x ∈ L  . The task of finding a verte x of a Delone ce ll of a point x , giv en a lattice ba sis of L , is called the closest vecto r pr oblem . Genera lly this is an NP-hard problem [DKS03]; ho wev er , there are algorith ms and implementations av ailable which can solv e this problem rather fast in lo w dimensio ns. The ortho gonal gr oup O( L ) of L is the group of all orthogon al transformatio ns A ∈ O( R n ) fixing L , i.e. A ( L ) = L . T he isometry gr oup Iso( L ) of L is th e group genera ted by O( L ) and all lattice translatio ns t v : R n → R n with t v ( x ) = x + v for v ∈ L . W e say th at two vertic es x and x ′ of V ( L ) are equ i v alent if th ere is a n A ∈ O( L ) so that A ( x ) = x ′ . Correspond ingly , we s ay tha t t wo Delone cell s D ( x ) and D ( x ′ ) are equi vale nt if there is an A ∈ Iso( L ) so that A ( D ( x )) = D ( x ′ ) . 3.2. M ain algorithm. Our algorith m finds a complete list of inequi v alen t full- dimensio nal Delone cells of L with respect to Iso( L ) . The enumeratio n process is a graph tra versal algorithm of the grap h of equiv alence classes of full-dimen sional Delone cells of L . T wo equi v alenc e classes are adjacent whene ver there is a face t between two of its representa ti ves . Note that this graph can hav e loops and multiple edges. For th e graph trav ersa l algorithm below one nee ds four subalgori thms, which we e xplain in the follo wing sections. Input: n , B ∈ Q n × n matrix of rank n . Output: Set M of a ll inequi v alent full-dimensio nal Delone cells of the lattic e B Z n with respect to the group Iso( B Z n ) . x ← an initial vertex of V( B Z n ) . (Section 3.3) T ← { D ( x ) } . M ← ∅ . while there is a D ∈ T do M ← M ∪ { D } . T ← T \ { D } . F ← facets of D . (Section 3.4) f or F ∈ F do D ′ ← full-dimension al Delone cell with F = D ∩ D ′ . (Section 3.4) if D ′ is not equi valen t to a Delone cell in M ∪ T then (Section 3.5) T ← T ∪ { D ′ } . end if end for end wh ile COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 9 1 b b 2 1 b b 2 x x c c v F I G U R E 1 . Finding an initial vert ex of V( L ) T wo full-dimens ional D elone cells D ( x ) and v + D ( x ) , both contain ing the origin , are equiv alent under O( L ) if and only if 0 and − v are equi valen t unde r the stabili zer g roup of D ( x ) in Iso( L ) . As a c onseq uence, we can compute th e vertice s of V ( L ) under O( L ) in the follo wing way: For ev ery orbit of full-dimen sional Delone cells giv en by a represe ntati v e D ( x ) , we compute the orbits of vertices of D ( x ) under the stabilizer group and get the correspond ing or bits of ver tices of V( L ) under O( L ) . 3.3. F inding an in itial verte x. Now we e xpla in a method for computin g an initial ver tex of t he V orono i cell of a lattice, i.e. a full- dimensio nal Delone cell containin g the o rigin. The metho d we p ropose is a so-called cuttin g-plan e algorit hm, which is a well-kno wn techniq ue in combinatori al optimization. Let us describe the geometric idea. W e start with an outer approximati on of the V oronoi cell giv en by linear inequalit ies. The first outer approxima tion is the polyto pe defined b y the inequalit ies ± b i · x ≤ 1 2 b i · b i for giv en latt ice basi s v ectors b 1 , . . . , b n . T hen w e fi nd a verte x x of the appro ximation by linear programming (see e.g. [S ch86]). D ecidin g whether the v erte x x belong s to the V oronoi cell V( L ) can be done a s f ollo ws: C ompute the vertices of the Delone cell D ( x ) . If t he origin is a v erte x of D ( x ) , then x is a v ertex of V( L ) . Otherwise x is not conta ined in V( L ) , and for all vertices v of D ( x ) w e hav e the strict inequali ty k x − v k < k v k . So the ne w linear inequ alities v · x ≤ 1 2 v · v together w ith the old ones pro vide a tighte r outer approximatio n of the V oronoi cell. Since we started with a compact outer approximatio n, finitely many iteratio ns of these steps suffice to find a vert ex of the V oronoi cell. One adv antage of th is method is that t he computation o f all f acets of the V oronoi cell is not required, i.e. we do not use V oronoi’ s characteriz ation (see Section 2.4) of facet defining vectors , which in volv es sol ving exponen tially many closest v ector proble ms. Figure 1 ill ustrate s this algorithm. Input: n , B = ( b 1 , . . . , b n ) ∈ Q n × n matrix of rank n . Output: verte x x of V ( B Z n ) . c ← random vector in Q n . B ← {± b 1 , . . . , ± b n } . do x ← a v erte x of the polytope { x : b · x ≤ 1 2 b · b for all b ∈ B } , which maximiz es c · x . 10 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN F c v F c v D D D’ F I G U R E 2 . Finding D ′ , th e full-di mensiona l D elone cell ad jacent to D at F E ← set of closest lattice vector s in B Z n to x . if 0 ∈ E then return x . end if B ← B ∪ E . end do 3.4. C omputing facets of , and finding adjac ent Delone cells. W e wan t to de- termine the f acets of a ful l-dimensi onal Delone ce ll, which is giv en by its verte x set. This represen tation con version pro blem can be solv ed by many diffe rent meth- ods. For details and implementations we refer to cdd [Fuk95], l rs [A vi93], pd [Mar97] and porta [CL97]. In order to explo it the symmetries we use the adjacency decompo sition method (see [CR96, BDS07, DSV07]). It a llo ws to compu te a complete lis t o f i nequi valen t face t represent ati ve s: W e compute an initial facet by linear programming and insert it into the list of orbit representa ti ves of facets. From any such orbit, we compute the list of facets adjacent to a represe ntati ve and insert it, if necessary , into the list of representat i ves un til all orbits ha ve been treate d. Computing adjacent facet s is itself a re presen tation con version problem in one dimensio n lower . So this method can be applied recursi vel y (see [BDS07, DSV07]). Note that our main algorithm is itself an adjac enc y decomposit ion method. After the computation of facets, we can compute adjacent full-dimensi onal De- lone cells: W e take an initia l v erte x v and so get a tentati v e empt y sphere. If the sphere is not empty , then we find another v erte x v and iterat e until the sphere is indeed empty . Figur e 2 illustrate s this algorithm. Input: n , B ∈ Q n × n matrix of rank n , a full-di mensiona l Delone cell D and a face t F of D . Output: verte x set V ′ of a full-dimen sional Delone cell D ′ with D ∩ D ′ = F . φ ← affine func tion on R n with F = { x ∈ D : φ ( x ) = 0 } and φ ( x ) > 0 on D − F . V F ← vertices of D belongin g to F . v ← a po int of B Z n with φ ( v ) < 0 . do COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 11 S ( c, r ) ← sp here around V F ∪ { v } . V ′ ← closest vectors in B Z n to c . if k v ′ − c k = r for a v ′ ∈ V ′ then return V ′ . end if v ← one ele ment of V ′ . end do One wa y to sp eed up the con ver genc e of this al gorith m in prac tice is to heuristi- cally choo se an initial vect or v w ith a sphere S ( c, r ) of small radius. 3.5. C hecking equiv alence. W e ha ve to test equi v alence and compute stabilize rs under the grou p Iso( L ) of Delone cells of diff erent dimensi ons. B elo w we propo se three dif ferent methods for this. W e can encode a Delone cell D by the center c ( D ) of the empty sphere around it or by the verte x barycenter g ( D ) = 1 | v ert D | P v ∈ vert D v of its vertex set v ert D . Both c ( D ) and g ( D ) are in v arian t u nder the stabilizer of D . Any two full-dimensi onal Delone cells D , D ′ are equal if and only if c ( D ) = c ( D ′ ) . Howe v er , it is possible if n ≥ 3 that c ( D ) lie s out side or on the bou ndary of D . If c ( D ) lies o n the b ound- ary of D then a facet containing c ( D ) , which is itself a Delone cell, has the same center as D . Hence, the sphere centers can be used to disting uish full-dimen sional Delone cel ls b ut the y do no t dist inguis h Delone cells. Therefore, we us e the v erte x baryce nter . In the first method we consider th e classes of the vert ex baryce nters g ( D ) and g ( D ′ ) in the quotient R n / L and check their equi v alence under the finite group Iso( L ) / L ≃ O ( L ) . T he generic methods u nderlyi ng isomorphism and stab ilizer computa tions genera te the full orbit of g ( D ) under Iso( L ) / L . This is typically memory intensi v e. In some cases we can use a method from comput ational group theory , which we no w exp lain in an example. Suppose g ( D ) is express ed in a basis ( b 1 , . . . , b n ) of L as ( α 1 2 · 3 , . . . , α n 2 · 3 ) with 0 ≤ α i ≤ 5 and we want to compute its sta bilizer unde r the group Iso( L ) / L . The ve ctor 2 g ( D ) reduc ed modulo L is exp ressed as ( ˜ α 1 3 , . . . , ˜ α n 3 ) with 0 ≤ ˜ α i ≤ 2 . W e fi rst compute the sta bilizer H of the vecto r 2 g ( D ) unde r the actio n of Iso( L ) / L . The stabilize r of g ( D ) und er Iso( L ) / L is equal to the stab ilizer of g ( D ) und er H . This method generalizes to more than two prime fa ctors an d it is more memory ef ficient because the ge nerate d orbits are smaller . The second m ethod uses finite metric spaces of t he verte x set of full-dimension al Delone cells obtained from the metric k v − v ′ k 2 [DL97, Chapter 14]. A finite metric space defines an edg e-weight ed graph. T esting if two edg e weighted graphs are isomorph ic can be redu ced to testing if two verte x-weigh ted graphs are isomorphi c (see [McK06, Page 25]). In practice, the program n auty [McK06] can solve the isomorph ism problem if the numb er of ve rtices of D and D ′ is not too large. If the metric spaces are not isomorphic , then D and D ′ are not equiv alent under Iso( L ) . If the y are isomorphic then ev ery graph isomorphism correspon ds to a linear isometry between D and D ′ [BDS07, DS V07]. For each of tho se isomorphisms, we check if it belongs to Iso( L ) . This meth od is u seful when the isometry g roup of D is small. 12 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN For the third method, we use lamination s o ver the n -dimensio nal lattice L . Let D be a Delo ne cell o f L with verte x barycen ter g . One defines an ( n + 1) - dimensio nal lattic e L ( g ) by embedding L ⊆ R n into R n +1 and adding layer s to it L ( g ) = { αv + h : h ∈ L, α ∈ Z } ⊆ R n +1 , where v ∈ R n +1 is cho sen so that v + g is orthog onal to the space spanned by L and normalized so that k v + g k = 1 . A varian t of this cons tructio n is used for exa mple to b uild the laminated lattic es, see [CS99, Chapter 6]. If φ is an element of O( L ( g )) pr eservin g eve ry layer of the lamination, th en it maps the vecto r v to some ve ctor w = v + h with h ∈ L . The function x 7→ φ ( x ) + h preserv es th e Delone cell and ev ery element preserving the layers is obtained in this way . In practic e, we can use the program AUT O (see [PP97]) of the packag e CARAT (see [OPS98]) for computing this automorphi sm group . The isomorphism pro blem is treated similarly using the prog ram I SOM . 3.6. W orking with Gram matrices and periodic structures . In many cases, it is more con ve nient to work with th e Gram matrix B T B inste ad of the lattice basis B (see [CS99, Chapter 2.2]). For insta nce, when B is irrational but B T B is rationa l. Note that our algorithms can be re formulate d in terms of Gram matrices. Note also that all our algor ithms can be modified to deal w ith periodic point sets, that is for finite unio ns of lattice transla tes. Our impleme ntation is av aila ble from [Dut08]. 3.7. C omparison. In [VB96] V iterb o and Biglier descr ibe another algorith m fo r computin g the V oronoi cell of a lat tice, called th e diamond cutting algorithm. As in our approa ch they start w ith a parallelep iped P defined by the basis vecto rs. Then the y determine all lattice vector s which lie in a sphere conta ining 2 P . This set contai ns all facet defining la ttice v ector s of V( L ) . S ucces si vel y they add cutting planes obtain ed from these vectors and update the complete face lattice of the ten- tati v e V oronoi cell. They terminate when its vo lume coincides with det L . Their implementa tion uses fl oatin g point arith metic. In comparison, our a pproac h has the follo w ing adv antages: W e use the presence of symmetry in an ef ficient way . W e do not need to compute a huge initial list of potent ial fa cet defining lat tice v ecto rs. Our algorithm d oes n ot need to compute the face lattice, not ev en fo r c omputin g the quan tizer const ant as explaine d belo w . Our implementa tion uses rational arithmeti c only . 4. C O M P U T I N G Q UA N T I Z E R C O N S T A N T S Recall from the introdu ction that the quanti zer constant of a lattice L is the inte gral G ( L ) = (det L ) − (1+2 /n ) Z V( L ) k x k 2 dx. A stan dard method for computing the integ ral G ( L ) is to decompos e V( L ) into simplice s. S uppos e that S is a simple x wit h v ertices v 1 , . . . , v n +1 in R n . Then (see COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 13 [CS99, Chapte r 21, T heore m 2]) the follo w ing holds: Z S k x k 2 dx = v ol S ( n + 1)( n + 2)        n +1 X i =1 v i      2 + n +1 X i =1 k v i k 2   . Thus G ( L ) can be obtained by summing the integrals of all simplices in a decom- positi on of V ( L ) . Sev eral practi cal methods for decomposin g a polytop e into sim- plices are discussed in [BEF98]. In our implementation , we use the triang ulatio n obtain ed by the program lr s . Ho we ver , this m ethod as well as the other methods exp lained in [BEF98] are sometimes impractica l and they do not use symmetri es. In order t o get a group in var iant decompositi on, we can use the barycen tric sub- di visio n of P . That is, giv en any flag F 0 ⊂ F 1 ⊂ . . . ⊂ F n = P of face s of P , we asso ciate the simp lex with verte x set g 0 , g 1 , . . . , g n where g i is the verte x baryce nter of F i . Note that in general there is a dif ference between the barycent er 1 v ol P R P xdx of a polytope P and its ve rtex ba rycent er 1 | vert P | P v ∈ vert P v . The group acts o n the baryce ntric subdi vision and the stabiliz er of each simplex is t ri v- ial. In practice , the number of orbits of flags can be too larg e. W e pro pose a hybr id app roach, which combines the benefits of bo th metho ds. Let F be the face t set of an n -dimensional po lytope P . W e can assume witho ut loss of gene rality that P has the origin as its verte x barycent er . W e the n hav e (4) Z P k x k 2 dx = X F ∈F Z con v( F , 0) k x k 2 dx. T o compute this sum, it is sufficient to compute the inte grals only for orbit repre- sentat iv es of facets . Let F be a facet of P and p F be a point in the affine space spann ed by F . Then w e can transf orm the integra l over the con e conv( F , 0) in the follo w ing way: Z con v( F , 0) k x k 2 dx = 1 n + 2  Z F k y − p F k 2 dy +2 Z F ( y − p F ) · p F dy + vo l F k p F k 2  . If p F is the ort hogon al projectio n of the origin 0 onto F then the second summand v anishe s. T his po int may not be in varian t under the a utomorph ism group of th e face t F , but the vertex baryce nter is. If w e use the vert ex barycen ter , we also ha ve to co mpute th e ba rycent er of th e p olytop e F as well as the v olume and the square integral . In order to use symmetries coming from non-o rthogo nal linear transfo rmations of P , we use the matrix v alued inte gral I 0 , 1 , 2 ( P ) = Z P  1 x  (1 , x t ) dx. This inte gral splits accordin g to I 0 , 1 , 2 ( P ) =  I 0 ( P ) I 1 ( P ) t I 1 ( P ) I 2 ( P )  , 14 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN where I 0 ( P ) = Z P dx = v ol P , I 1 ( P ) = Z P x dx, I 2 ( P ) = Z P xx t dx. Let G be a group of automorph isms of P . If g ∈ G acts on R n as x 7→ Ax + v then we define H ( g ) =  1 0 v A  the corresp ondin g ( n + 1) × ( n + 1) matri x acting on homogene ous coordina tes. Let O 1 , . . . , O r be the G -orbits of facets of P , with repres entati ves F 1 , . . . , F r . Then the integral I 0 , 1 , 2 ( P ) simpl ifies to I 0 , 1 , 2 ( P ) = r X i =1 | O i |   1 | G | X g ∈ G H ( g ) I 0 , 1 , 2 (con v ( F i , 0)) H ( g ) t   . Assume that I 0 , 1 , 2 (con v ( F i , 0)) is already computed. T o compute the sum in the parent hesis, we first incrementall y compute a basis of the affine hull of the orbit { H ( g ) I 0 , 1 , 2 (con v ( F i , 0)) H ( g ) t : g ∈ G } . The on ly G -in v ariant element of the af fine hull is the sum we wan t to compute. W e no w want t o compu te I 0 , 1 , 2 (con v ( F, 0)) in terms of lower dimensional inte- grals. The i nteg ral depend s on the chosen basis. If f is an affine transfo rmation of R n , then the change of v ariabl es formula for integ rals giv es H ( f ) I 0 , 1 , 2 ( f P ) H ( f ) t | det H ( f ) | = I 0 , 1 , 2 ( P ) , for any an n -dimensiona l polytop e P in R n . This allo ws to compute I 0 , 1 , 2 ( P ) for anothe r basis. So, we can cho ose a coordinat e system such that F =  1 x  : x ∈ F ′  ⊂ R n , where F ′ ⊂ R n − 1 is an ( n − 1) -dimensi onal polyto pe. W e th en ha ve the follo wing formulas : I 0 (con v ( F, 0)) = 1 n I 0 ( F ′ ) , I 1 (con v ( F , 0)) = 1 n +1  I 0 ( F ′ ) I 1 ( F ′ )  , I 2 (con v ( F , 0)) = 1 n +2 I 0 , 1 , 2 ( F ′ ) . For computing I 0 , 1 , 2 ( F ′ ) , we ha v e two optio ns: Either we use the first method of this secti on, which in vol ves computing a triangula tion or we apply the abo ve method recursi vely . The de cision is made heurist ically , dependin g on the size of the automorphism gro up of F and it s nu mber of vertic es. In order to reduc e the size of the comput ation, one can store interme diate results. Those metho ds are genera l and apply to any polyto pe and any p olynomia l func- tion, which we want to inte grate ove r P . Note that a similar method of using the standa rd formula (4) has been used for computing the v olume in [BEF98] under the name of Lasserr e’ s method ([Las98]), albei t in a non-g roup setting. COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 15 5. R E S U L T S In this section, we collect results from our implementa tion of the algorithms exp lained in Sections 3 and 4 . W e obta in previo usly unkno w n exact cov ering den- sities an d quant izing constan ts of se vera l prominent lattic es and their d uals. Recall that the dual L ∗ of a lattice L ⊂ R n is defined by L ∗ = { x ∈ R n : y · x ∈ Z for all y ∈ L } . The cov ering dens ity of an n -dimensi onal lattice L is µ ( L ) n det L v ol B n , where B n is the unit ball in R n . Other c omputati ons of V oronoi cells of lattic es can be found in [CS91], [ EMS03, Chapter 5] and [MP95] . All computations are done in exac t rational arith metic. In the tables the cover ing densi ties are giv en in floating poin t; the exa ct expres sions would be too lar ge. 5.1. C oxeter lattices. The root latti ce A n is defined by A n = ( x ∈ Z n +1 : n +1 X i =1 x i = 0 ) . If r di vides n + 1 , the Coxeter lattic e A r n (see [Cox51]) is defined by translat es of A n : A r n = A n ∪ ( v r n + A n ) ∪ . . . ∪ (( r − 1) v r n + A n ) , where v r n = 1 n +1 P n +1 i =2 ( e i − e 1 ) . The dual lattice of A r n is A n +1 /r n . The D elone decompositi on of the lattice A r n has b een s tudied i n [ Anz02, Anz 06, Bar94] up to dimensi on n = 15 , hoping to obtain lattices with low cove ring densit y . One plea sant fac t is that the symmetry group of A r n contai ns the group Sym( n + 1) × Z 2 . Latter can be represented as a permutation group acting on n + 3 points, which drastically simplifies isomorp hism computations . In T able 1 we list the obtaine d results. Note that the lattices A 6 17 , A 10 19 , A 7 20 and A 11 21 turn out to giv e ne w record sphere cover ings. Up to dimension 8 all those lat- tices are w ell kno wn and their V oronoi cells can be ob tained by stand ard computer algebr a software. Our list is complete up to dimension 21 . For the missing cases we coul d not fi nish the computa tion. 5.2. L aminated lattices. The laminated lattices , whic h are defined in [CS99, Chap- ter 6], giv e the best kno wn lattice sphere packin gs in many dimensions. The Delone subdi vision is kno wn up to dimensio n 8 and in dimension 24 for all laminated lat- tices and their du als [CS 99, Chapters 2 1, 23 , 25]. In dime nsion 16 , the cov ering densit y of Λ 16 is kno w n [CS99, Chapter 6]. In T able 2 we li st the obtained results, which are complete up to dimension 17 . 16 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN lattice # orbits cov erin g densit y lattice # orb its cov ering density A 2 9 6 18 . 54 3333 A 5 9 5 4 . 34018 4 A 2 11 6 94 . 09 0996 A 3 11 11 27 . 08 9662 A 4 11 16 5 . 59833 7 A 6 11 4 7 . 61855 8 A 2 13 10 134 . 62348 4 A 7 13 10 7 . 86 4060 A 3 14 17 32 . 31 3517 A 5 14 31 9 . 00 6610 A 2 15 10 722 . 45264 2 A 4 15 19 25 . 36 3859 A 8 15 10 11 . 60 1626 A 2 17 15 1068 . 51 3081 A 3 17 26 240 . 51158 0 A 6 17 73 12 . 35 7468 A 9 17 24 17 . 23 1927 A 2 19 15 5921 . 05 6764 A 4 19 58 40 . 44 5924 A 5 19 80 25 . 60 9662 A 10 19 80 21 . 22 9200 A 3 20 40 307 . 20948 7 A 7 20 187 20 . 36 6828 A 2 21 21 8937 . 56 7486 A 11 21 64 27 . 77 3140 A 3 23 55 2405 . 03 2746 A 4 23 85 205 . 56122 5 A 6 23 187 79 . 57 5330 A 8 23 495 31 . 85 8162 A 12 23 100 43 . 23 1587 A 5 24 144 115 . 0115 91 A 13 25 210 54 . 47 2182 A 3 26 75 3 184 . 1 387034 A 9 26 1231 50 . 937168 A 4 27 156 350 . 1370 31 A 7 27 650 81 . 86 9181 A 14 27 338 76 . 90 9712 A 3 29 102 25664 . 644103 A 5 29 347 202 . 0403 31 A 6 29 711 154 . 3298 31 A 10 29 3581 84 . 324725 A 15 29 678 114 . 0842 19 A 16 31 1225 33 . 934941 T A B L E 1 . Number of orbits of full-dimension al Delone cells and cov erin g density for some Coxete r lattices 5.3. S horter Leech lattice . The 4600 shortest vectors of Λ ∗ 23 of define a sublattice of inde x 2 , called the shorter Leec h lat tice O 23 ([CS99, Page 179, 420, 441]) . The Delone decomp osition (see T able 3 ) is remarka ble in many respec ts: There are only 5 orbits and the first one has the full symmetry group o f the lattice. It turns COMPLEXITY AND ALGORITHMS FOR COMPUTING V OR ONOI CELL S OF LA TT ICES 17 lattice # orbits cov erin g densit y lattice # orbi ts cov ering density Λ 9 5 9 . 00352 7 Λ ∗ 9 9 9 . 00352 7 Λ 10 7 12 . 40 8839 Λ ∗ 10 21 9 . 30 6629 Λ max 11 11 24 . 78 1167 Λ max ∗ 11 18 19 . 24 3468 Λ min 11 18 24 . 78 1167 Λ min ∗ 11 153 8 . 17043 2 Λ max 12 5 30 . 41 8954 Λ max ∗ 12 8 42 . 72840 8 Λ mid 12 23 30 . 41 8954 Λ mid ∗ 12 52 19 . 17 6309 Λ min 12 13 30 . 41 8954 Λ min ∗ 12 78 12 . 29 2973 Λ max 13 18 60 . 45 5139 Λ max ∗ 13 57 43 . 21 4494 Λ mid 13 46 35 . 93 1846 Λ mid ∗ 13 125 19 . 15 5991 Λ min 13 129 60 . 45 5139 Λ min ∗ 13 5683 13 . 724864 Λ 14 65 98 . 87 5610 Λ ∗ 14 1392 34 . 721750 Λ 15 27 202 . 91087 3 Λ ∗ 15 108 25 . 64 2067 Λ 16 4 96 . 50 0266 Λ ∗ 16 4 96 . 50026 6 Λ 17 28 197 . 71949 9 Λ ∗ 17 720 100 . 1731 01 Λ 18 239 301 . 1923 34 Λ 23 709 7609 . 031 33 T A B L E 2 . Number of orbits of full-dimension al Delone cells and cov erin g density for some laminated lattices and their duals number of vert ices size of stabilize r group 94208 84610 84262 4000 32 1344 24 10200 960 24 1320 24 1320 T A B L E 3 . Orbits of full-dimens ional Delone cells of O 23 out th at Λ ∗ 23 = O 23 ∪ ( v + O 23 ) where v is the cen ter of a centrally symmetric Delone cell lying in the first orbit . The co ver ing density of O 23 is 15218 . 0 62669 . 5.4. C ut lattices. The cut polytop e CUT n is a famous polytope appearing in com- binato rial optimization (see [DL97]). It has 2 n − 1 ver tices and is of dimensio n n ( n − 1) 2 . T he lattice generated by its vertices is called cut lattice and is denoted by L (CUT n ) (see [DG 95]). The polytop e CUT n is one of its full-d imension al Delone cells. W e list our results in T able 4. 18 MA THIEU DUTOUR SIKIRI ´ C, A CHILL SCH ¨ URMANN, AND FRANK V ALL ENTIN lattice dimensio n # orbits cov ering density L (CUT 3 ) 3 2 2 . 094 39 L (CUT 4 ) 6 4 5 . 16771 L (CUT 5 ) 10 12 40 . 8 0262 L (CUT 6 ) 15 112 255 . 425 5 T A B L E 4 . Dimensions, number of orbits of full-dimensio nal De- lone cells and cov ering densit y of some cut lattice s lattice quanti zer constan t Λ 9 151301 2099520 ≈ 0 . 07206 Λ ∗ 9 13715142 91 19110297 600 ≈ 0 . 07176 A 2 9 2120743 9 √ 5 . 2 8 13271040 ≈ 0 . 07216 6 A 5 9 86514275 63 9 √ 2 . 5 8 26578125 000 ≈ 0 . 07207 9 D + 10 4568341 64512000 ≈ 0 . 07081 A 2 11 452059 11 √ 35702400 ≈ 0 . 07174 A 3 11 28754428 1699 11 √ 4 . 3 10 13258390 06800 ≈ 0 . 07042 6 A 4 11 63876579 54959 11 √ 3 . 2 9 46506442 752000 ≈ 0 . 07049 4 D + 12 29183629 41277600 0 ≈ 0 . 07070 0 K 12 79736194 1 √ 36567561 000 ≈ 0 . 07009 5 T A B L E 5 . Quantize r constants of some lattices 5.5. Q uantizer constants. 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