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Normalized second moments of the lattice quantizers
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Numbers
family
$n$ 
$G(L)$
$\mathbb{Z}^n$
1:
1/12
comment: The interval has $G(\mathbb{Z})=1/12$.
$\mathbb{Z}^n$
2:
1/12
comment: The square lattice.
$\mathbb{Z}^n$
3:
1/12
comment: The simple cubic lattice.
$\mathbb{Z}^n$
4:
1/12
$\mathbb{Z}^n$
5:
1/12
$\mathbb{Z}^n$
6:
1/12
$\mathbb{Z}^n$
7:
1/12
$\mathbb{Z}^n$
8:
1/12
$\mathbb{Z}^n$
9:
1/12
$\mathbb{Z}^n$
10:
1/12
$\mathbb{Z}^n$
11:
1/12
$\mathbb{Z}^n$
12:
1/12
$\mathbb{Z}^n$
13:
1/12
$\mathbb{Z}^n$
14:
1/12
$\mathbb{Z}^n$
15:
1/12
$\mathbb{Z}^n$
16:
1/12
$\mathbb{Z}^n$
17:
1/12
$\mathbb{Z}^n$
18:
1/12
$\mathbb{Z}^n$
19:
1/12
$\mathbb{Z}^n$
20:
1/12
$\mathbb{Z}^n$
21:
1/12
$\mathbb{Z}^n$
22:
1/12
$\mathbb{Z}^n$
23:
1/12
$\mathbb{Z}^n$
24:
1/12
$A_n$
1:
1/12
comment: This one-dimensional lattice is similar to $\mathbb{Z}$.
$A_n$
2:
0.08018753738744802284849288618082742439550024323196206611369476756721912115318518690190491605357632295
comment: The hexagonal lattice, with $G(A_2)=5/(36\sqrt{3})$.
$A_n$
3:
0.07874506561842957279795066295488927191064071654384424875512344450970622978212246773308728515226594338
comment: The face-centered cubic lattice.
$A_n$
4:
0.07801970224724923613371052521350892591193743902612238255649806953402308775582538836786950243862055390
$A_n$
5:
0.07764745764128658280366412253015776038964359072998404403244620629645437687688795030292236305590817516
$A_n$
6:
0.07746643139994469031301884563684811662980480583501311624493694208488062871442276405902205966154914126
$A_n$
7:
0.07739553589254939712499235868305169824602663330401613194565120818493702387190396877585406219225174538
$A_n$
8:
0.07739067168673627796891727089999998786448618460889271763633495546947726448964370358831959959088341414
$A_n$
9:
0.07742636826811270597266794515369692782143337437397447352913919774090861419927038908568873884471090393
$A_n$
10:
0.07748723294362150803726328055355981443808246015320561849201604534646575224424905845795022009564025374
$A_n$
11:
0.07756363564201440762016174157948302088651015677251531260435642302463350498957603729370718813326072255
$A_n$
12:
0.07764938994141730498519385291906453507952865263482072081946254475716839900940359453202988154801631804
$A_n$
13:
0.07774044206387245858511468661979250424337782807797854095244868747366528929414427142818774569969672179
$A_n$
14:
0.07783409816522920596806578912649443854935585792846278936908368970019102576431405756306830897564760735
$A_n$
15:
0.07792855276338635419149753205198426748908205968690065164780558357299383915658670187639578579437754324
$A_n$
16:
0.07802259235293355458555003420172640814050662395312738234621414772949697164838984881578650139161951258
$A_n$
17:
0.07811540439275903321447820541113176657815389444411000607288300106679742516249642567716151129060077712
$A_n$
18:
0.07820645152469503624872897293892310436463361822131332337591524483521945396097853614275847397422515481
$A_n$
19:
0.07829538719637768410510811915051660393623884114949669025604572459815554721419603318601359528633284029
$A_n$
20:
0.07838199814660318556707272701760556466075411822593124619377649517556487409584875646642734963069924270
$A_n$
21:
0.07846616465789947089614386940899289039842400038592841398364083193812015142458114031871989488185246481
$A_n$
22:
0.07854783276243637000503667933188460273028125122071982891395105766033317464119455832461354350956331767
$A_n$
23:
0.07862699461210796674510533244424338315184234365904974941960184073050781339210103975690178031309350056
$A_n$
24:
0.07870367449990510520722732916472428906969382622645897143574873982428350612414941479372802441241077652
$A_n^{*}$
3:
0.07854328121717653651115210194316108580062076362004481038113913925697685835119834451093057305203697444
comment: The body-centered cubic lattice, with $G(A_3^*)=19/(192\sqrt[3]{2})$.
$A_n^{*}$
4:
0.07755875678600730544049716116277537477285144609494943578746064953581886008699944506641532021336173560
$A_n^{*}$
5:
0.07692194906147078182058948105630438095146189233261519641398380482754142054570365264144095307043204872
$A_n^{*}$
6:
0.07648976119263003796597272023934071080637239140372901521938134075982140708233874235453571754382620943
$A_n^{*}$
7:
0.07618727598255273705007559514043497241671804048483062747611297521444487002399197465100456172710776852
$A_n^{*}$
8:
0.07597152745983311487746465140654163816352103219362984775837810621140668784242658678029364649617554015
$A_n^{*}$
9:
0.07581616958113530888838546777351085113542858624452039442375036312855605261006345008571109323000067352
$A_n^{*}$
10:
0.07570419763229146999509868763333271450677754083782952232495403974593136066175695088384253465554992629
$A_n^{*}$
11:
0.07562418172023446363838989602597120012927611075864826157672736457104886837459547461110282700212485493
$A_n^{*}$
12:
0.07556818213655412115141283213272530536171258560413646993706107274750426675688442206093264123895867161
$A_n^{*}$
13:
0.07553053150079142906535472549963684566360187267649403666086521517522094478587452864799435227888640973
$A_n^{*}$
14:
0.07550709093228317717488514616682717058829207984352207061964607843928063326370485731075806734879889386
$A_n^{*}$
15:
0.07549477849938701645896498012670610145312115232135546129611075860599256327497646382455471046035836265
$A_n^{*}$
16:
0.07549126065547157134786068863382864482208705911494303259124846204291384969540090804772768176880654176
$A_n^{*}$
17:
0.07549474474112937386976563368271388207661017121563956685431485467295341968473325083092212095783438409
$A_n^{*}$
18:
0.07550383609567527460865211394589853800349396706768801955158744971811223159636015036868953524286715431
$A_n^{*}$
19:
0.07551743758584744882726041623274144144931475723821452968403398942796509800223275272229656886461192540
$A_n^{*}$
20:
0.07553467764290796192961790537746266620001029172542851160627076862460530203922534211099804802427406817
$A_n^{*}$
21:
0.07555485786321693891833855345322219061570632040511809665195789699803916871090352407693396715597271435
$A_n^{*}$
22:
0.07557741428617642619694442778127065169610805507571933158234000442518150955230919728350655779605397219
$A_n^{*}$
23:
0.07560188839581792057693940958336046507368155616543863169923632814037492206499288110313022933539089549
$A_n^{*}$
24:
0.07562790514063651774269235525382711921089762594457110660899303227687487120494163983409900849152472834
$D_n$
4:
0.07660323462854264847675813922802531258919055991625135396373681914116467591669746460439599144274725186
comment: The Voronoi cell is the regular $24$-cell.
$D_n$
5:
0.07578582832551990411736299006532226193406417714890708210187526210487268386010291388205880674247158953
$D_n$
6:
0.07559052628419997498817645901296706001864253942380252427658503627739126217258186539397829586962956938
$D_n$
7:
0.07568570248879992817344888443122731412718981008108152179248495898317106384977118531803170633314192840
$D_n$
8:
0.07591425971040478513475438327105412246889198201832082389286068831718070704444950776101536149021390952
$D_n$
9:
0.07619946514249536274515867159253653562273069867900108514300800481303014842246271906614423857763272430
$D_n$
10:
0.07650292828965939404530857729367465718301405244270329817883008329301742580249049180443328539781346094
$D_n$
11:
0.07680529873071396745835805022833884687536482596883264444622817690859252575450501079772345874293834002
$D_n$
12:
0.07709700445445243983328880625302512088350878406637926567744050894487862721001790801670896062745206517
$D_n$
13:
0.07737361857177741407161428655293800019418669517736921329792907212767421821110936365916490263362474987
$D_n$
14:
0.07763345693690628613664338913272313037432393267925177329588926840437906862687681459612266233351735246
$D_n$
15:
0.07787629589768101900095897545898013886232230731792421619667594557253585565315556899620956527155423843
$D_n$
16:
0.07810267279745667966443154514484440900574330155313769927759866889712021907230777171260030822543614191
$D_n$
17:
0.07831349890347222345662969637624054018673077333278398318335820294701056618987605609995955866340068654
$D_n$
18:
0.07850984402436088434776619257989242438883898663181690326826465151798876974948051438270385022558863939
$D_n$
19:
0.07869281733185193519512896962484792968346623332314173833988405437599242403048400123597754158784656471
$D_n$
20:
0.07886350285608729349366115701981654030825396127256675804992299329697998624053572842840860787026168722
$D_n$
21:
0.07902292631101859445754300583761912844626428465347867139550897335945083048456157861232004380955512609
$D_n$
22:
0.07917203989769328908887461908084333511529810536971575777191520916660396440125917165525907458161335715
$D_n$
23:
0.07931171734406418259877866925579826896974304371312300471608295119358608638540262377697201639227969426
$D_n$
24:
0.07944275465070920263402769068618414331397033219767752015662886614512244605134114750630420326306032378
$D_n^{*}$
5:
0.07562544254234404815552819457503210918709617326049953858196263865417959039799888264242587615490818220
$D_n^{*}$
6:
0.07512029275662884643106087053315627367983344546484189444734792087354514936358032239299279107382830472
$D_n^{*}$
7:
0.07485878914788694466990310739526423476597133910048387090651413558083029514224641092907083946935509651
$D_n^{*}$
8:
0.07473508668881463273394121149865424726702003022662410742917313088273654477765334499243046892138511257
$D_n^{*}$
9:
0.07469309658961008362372960477814285022002237610781495644200719401404340698886997400778544582238444485
$D_n^{*}$
10:
0.07470137910807957663529302875208890739239867678728879622631262160564552220942397604958654988282068417
$D_n^{*}$
11:
0.07474127041813927357006109886136802682534464668628048542433096454896270116809946038716274556237076054
$D_n^{*}$
12:
0.07480123806198952596957427176075966348372620659248429758061100709073260697494078621104528900319500351
$D_n^{*}$
13:
0.07487391898622613354133892525532549761822855711479802882486810886397634290764724122316577054632361323
$D_n^{*}$
14:
0.07495449203054999957231189797193171872590937471393591911084331169121113875727506144306282478581199850
$D_n^{*}$
15:
0.07503973797341055490921709057221025001493608895227189744558943270099989592419533289652792364255099853
$D_n^{*}$
16:
0.07512747582080452475822031816076167828765295441383984657597764138712041555067445670373185636975811780
$D_n^{*}$
17:
0.07521621332298647001036058381996414849955224277265042623308257757561673140527052041243306422484791876
$D_n^{*}$
18:
0.07530492416138779712774813263307430683841236512188652549519987745241137477896251560893769696780910701
$D_n^{*}$
19:
0.07539290241236745026027203054884331504556002202824104996102807936267564805085643045839315287608914552
$D_n^{*}$
20:
0.07547966547481905253578937689047101148055506689204815685775864996557016371275755745263353986999934864
$D_n^{*}$
21:
0.07556488813202524584313023138204507130413288046756492584863140169208029468991761203686724852130404682
$D_n^{*}$
22:
0.07564835704463569038550470774556669120516358662363692263073260803486107712491648098321239713294014488
$D_n^{*}$
23:
0.07572993890590397609932921231122698897051509243932159840173130234950911695774132525779252903529774104
$D_n^{*}$
24:
0.07580955788720755040874034709303054638389775474607323851640646603131098208205637972423600035724675048
$E_n$
6:
0.07434671229067895710792735379543176398768411141907757468662859296656600383864434938321657699988273178
$E_n$
7:
0.07323063356895673981639393766570526765633565408980346670850665943237145941076921747203700208543592854
$E_n$
8:
929/12960
comment: The exact value is $G(E_8)=929/12960$.
Definition
The normalized second moment is $G(L)=\frac{1}{n v_L^{1+2/n}}\int_{V_L}\|x\|^2\,dx$, where $L$ is an $n$-dimensional lattice, $V_L$ its Voronoi cell at the origin, and $v_L=\operatorname{vol}_n(V_L)$, [1].
Parameters
family
—   family of the lattice
$n$
—   dimension ($n\geq 1$ for $\mathbb{Z}^n$, $A_n$, $A_n^{*}$ and $\Lambda_n$; $n\geq 4$ for $D_n$ and $D_n^{*}$; $n\in\{6,7,8\}$ for $E_n$ and $E_n^{*}$; $n=12$ for $K_n$)
Formulas
(1)
$G(cUL)=G(L)$ for $c>0$ and every orthogonal map $U$.
(2)
$G(\mathbb Z^n)=1/12$ for $n\geq1$.
(3)
$G(A_n)=(n+1)^{-1/n}\left(\frac1{12}+\frac1{6(n+1)}\right)$ for $n\geq1$, [1], equation (32).
(4)
$G(D_n)=2^{-2/n}\left(\frac1{12}+\frac1{2n(n+1)}\right)$ for $n\geq4$, [1], equation (33).
(5)
$G(E_6)=\frac5{56\,3^{1/6}}$, $G(E_7)=\frac{163}{2016\,2^{1/7}}$, and $G(E_8)=\frac{929}{12960}$, [1], equations (34)–(36).
(6)
For $D_n^*=\mathbb Z^n\cup(\mathbb Z+\tfrac12)^n$, the Voronoi cell is $\{x:|x_i|\leq\tfrac12,\ \sum_{i=1}^{n}|x_i|\leq n/4\}$ and has volume $1/2$. Integrating $\|x\|^2$ over this cell gives $G(D_n^*)$, [1], Section III-F.
(7)
$G(L)=I(V_L)/(n v_L^{2/n})$, where $I(V_L)$ is the mean squared distance from the origin of a uniformly distributed point in $V_L$, [1].
Comments
(8)
The lattice names follow the classical-lattice packing table. The root lattices $A_n$, $D_n$, and $E_n$ have minimal squared length $2$, and $\mathbb Z^n$ has minimal squared length $1$. The star denotes the Euclidean dual. These conventions identify the same lattices as [2]. The value of $G(L)$ is unchanged by scaling the lattice.
(9)
For a point distributed uniformly in the Voronoi cell, $G(L)$ is its mean squared distance to the origin per dimension, divided by $v_L^{2/n}$. In [1], the symbol $I(V_L)$ denotes $v_L^{-1}\int_{V_L}\|x\|^2\,dx$, the mean squared distance to the origin.
(10)
Dual lattices need not have equal second moments. For example, $A_3$ is the face-centered cubic lattice and $A_3^*$ is the body-centered cubic lattice; their Voronoi cells are respectively a rhombic dodecahedron and a truncated octahedron, [1].
(11)
The Voronoi cell is $V_L=\{x\in\operatorname{span}_{\mathbb R}(L):\|x\|\leq\|x-v\|\text{ for every }v\in L\}$. Distances and volume use the Euclidean metric on the span of $L$.
(12)
The values for $A_1^*$, $A_2^*$ and $D_4^*$ are represented by $A_1$, $A_2$ and $D_4$, respectively, because each pair is similar. The lattice $E_8$ is self-dual, so its entry also represents $E_8^*$. The laminated lattices $\Lambda_1$ through $\Lambda_8$ are similar, respectively, to $A_1,A_2,A_3,D_4,D_5,E_6,E_7,E_8$ and are represented by those entries, [2].
Programs
(P1)
Sage
# Normalized second moment of the root lattice A_n.
n = 2
R = RealBallField(400)
G = R(QQ(1)/12 + QQ(1)/(6*(n+1))) / R(n+1)**(QQ(1)/n)
G  # 5/(36*sqrt(3))
References
[1]
J. H. Conway and N. J. A. Sloane, Voronoi regions of lattices, second moments of polytopes, and quantization, IEEE Transactions on Information Theory 28 (1982), 211–226. (doi) https://neilsloane.com/doc/Me82.pdf
Links
Similar tables
Packing densities and Hermite numbers of the classical lattices —   Packing invariants of the same lattice families.
Covering radii and covering densities of the classical lattices —   Covering invariants, which describe the largest distance to the lattice rather than the mean squared distance.
Values of the Epstein zeta function of the classical lattices —   Sums of inverse powers of vector norms for the same lattice normalisations.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $\mathbb Z^n$ and $A_n$ for $1\leq n\leq24$, $A_n^*$ for $3\leq n\leq24$, $D_n$ for $4\leq n\leq24$, $D_n^*$ for $5\leq n\leq24$, and $E_6,E_7,E_8$, giving 114 entries; dimension 24 is the common endpoint used for comparison with the classical-lattice packing and covering tables, and $E_6^*$, $E_7^*$, laminated lattices in dimensions $n\geq9$, and $K_{12}$ are outside this initial range)
How they were obtained:

All rational operations are exact. The closed forms for $A_n,D_n,E_n$ and the recurrence for $A_n^*$ come from [1]. For $D_n^*$, exact inclusion-exclusion integrates its clipped-cube Voronoi cell. Positive rational roots are evaluated in arb with 64 guard bits beyond the requested 100 significant digits. The largest measured relative radius at that working precision was less than $2\cdot10^{-119}$. The 26 rational entries, including every cubic lattice, $A_1$, and $E_8$, are written as exact fractions.

more

The attached checks independently integrate root-lattice affine Weyl alcoves, product-permutohedron facets, and a coordinate of the clipped cube against the distribution of the sum of the other coordinates. Exact nth powers agree for all 114 entries, and computations at 100 and 150 significant digits agree. Controls include a line segment, a triangle, one- and two-dimensional dual cubic cells, five named lattice values, and the seven permutohedron moments printed in [1]. Every stored value is also compared with these independent exact forms after readback.



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