Stored rules

Written by build/build_data.jl. q: the argument of ghpos(d, q) and lepos(d, q); p = 2q - 1: degree of exactness; n: number of nodes; ρ = n^(1/d) / q: the node count relative to the q^d Gauss product grid (1.00 is that grid, smaller is better); Möller: Möller's lower bound on n (bold n: bound attained, proven minimal); rel. err.: largest relative monomial error of the stored Float64 rule; origin: own, or the published rule it is, or descends from (see Credits).

GH — Gaussian weight

d = 2

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	4	1.00	4	2.2e-16	own
3	5	7	0.88	7	2.2e-16	same-rule: Stroud 1971, E_2^{r^2}:5-4 (identified 2026-09-11; same up to rotation, which the Gaussian weight permits)
4	7	12	0.87	12	3.6e-16	own
5	9	18	0.85	17	3.0e-16	transcribed: Haegemans and Piessens 1977
6	11	25	0.83	24	5.9e-16	transcribed: Haegemans and Piessens 1976, hexagonal
7	13	34	0.83	31	7.0e-16	transcribed: Cools and Haegemans 1988
8	15	44	0.83	40	6.5e-16	own
9	17	55	0.82	49	8.7e-16	own
10	19	68	0.82	60	5.5e-16	own
11	21	82	0.82	71	7.3e-16	own
12	23	97	0.82	84	7.1e-16	own
13	25	114	0.82	97	7.7e-16	own
14	27	132	0.82	112	1.0e-15	own
15	29	153	0.82	127	1.1e-15	own
16	31	178	0.83	144	6.2e-16	own
17	33	208	0.85	161	1.1e-15	own

d = 3

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	6	0.91	6	2.8e-17	own
3	5	13	0.78	13	2.2e-16	transcribed: Stroud 1971, icosahedral (closed form)
4	7	27	0.75	26	4.4e-16	same-rule: Stroud 1971, En^{r^2}:7-1 option 1 (identified 2026-09-11 vs Burkardt enr2071)
5	9	45	0.71	43	3.0e-16	same-rule: Konyaev 1977, rule 1 for the weight exp(-rho^2) (identified 2026-09-16 against the closed form printed in Dokl. Akad. Nauk SSSR 233 no. 5, 784-787; same up to rotation, node displacement 9.7e-16)
6	11	77	0.71	68	7.9e-16	own
7	13	128	0.72	99	4.9e-16	own
8	15	184	0.71	140	6.0e-16	own
9	17	264	0.71	189	6.5e-16	own
10	19	354	0.71	250	1.1e-15	own
11	21	476	0.71	321	6.1e-16	own
12	23	597	0.70	406	5.8e-16	own
13	25	776	0.71	503	5.5e-16	own
14	27	966	0.71	616	7.4e-16	own
15	29	1242	0.72	743	5.6e-16	own
16	31	1848	0.77	888	6.6e-16	own
17	33	2226	0.77	1049	7.8e-16	own

d = 4

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	8	0.84	8	2.2e-16	own
3	5	22	0.72	21	3.0e-16	own
4	7	49	0.66	48	4.4e-16	same-rule: Stroud 1971, En^{r^2}:7-1 option 1 (identified 2026-09-11 vs Burkardt enr2071)
5	9	116	0.66	91	5.9e-16	own
6	11	193	0.62	160	5.9e-16	own
7	13	414	0.64	259	4.4e-16	own
8	15	577	0.61	400	5.2e-16	own
9	17	1056	0.63	589	3.9e-16	own
10	19	1505	0.62	840	9.9e-16	own
11	21	2318	0.63	1161	6.5e-16	own
12	23	3238	0.63	1568	8.6e-16	own

d = 5

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	10	0.79	10	2.2e-16	same-rule: Stroud 1971, E_n^{r^2}:3-1 (identified 2026-09-11, displacement 0.0; order 2N = 10 at N=5)
3	5	32	0.67	31	3.0e-16	transcribed: Stroud and Secrest 1963, E5r2:5-1
4	7	83	0.61	80	3.0e-16	same-rule: Stroud 1971, En^{r^2}:7-1 (identified 2026-09-11, comparerules.jl over B_5)
5	9	244	0.60	171	3.9e-16	own
6	11	485	0.57	332	5.9e-16	own
7	13	1135	0.58	591	1.1e-15	own
8	15	1767	0.56	992	1.3e-15	own
9	17	3986	0.58	1581	2.8e-15	own
10	19	7174	0.59	2422	4.7e-15	own
11	21	13199	0.61	3583	5.1e-15	own

Le — uniform weight on the cube

d = 2

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	4	1.00	4	5.6e-17	own
3	5	7	0.88	7	5.6e-17	own
4	7	12	0.87	12	5.6e-17	own
5	9	17	0.82	17	5.6e-17	same-rule: Festa and Sommariva 2012, SMR09 (count first published by Moller 1976)
6	11	24	0.82	24	5.6e-17	own
7	13	33	0.82	31	2.2e-16	same-rule: Festa and Sommariva 2012, SMR13
8	15	43	0.82	40	1.1e-16	same-rule: Festa and Sommariva 2012, SMR15
9	17	54	0.82	49	1.1e-16	own
10	19	67	0.82	60	5.6e-17	own
11	21	81	0.82	71	5.6e-17	own
12	23	96	0.82	84	5.6e-17	own
13	25	113	0.82	97	5.6e-17	transcribed: Festa and Sommariva 2012, SMR25
14	27	131	0.82	112	5.6e-17	own
15	29	150	0.82	127	1.1e-16	own
16	31	171	0.82	144	2.2e-16	own
17	33	194	0.82	161	2.2e-16	own
18	35	216	0.82	180	2.2e-16	own
19	37	242	0.82	199	1.1e-16	own
20	39	268	0.82	220	1.1e-16	own
21	41	294	0.82	241	5.6e-17	own
22	43	324	0.82	264	1.1e-16	own
23	45	356	0.82	287	2.2e-16	own
24	47	388	0.82	312	1.1e-16	own
25	49	422	0.82	337	5.6e-17	own
26	51	456	0.82	364	5.6e-17	own
27	53	492	0.82	391	5.6e-17	own
28	55	528	0.82	420	5.6e-17	own
29	57	570	0.82	449	5.6e-17	own
30	59	606	0.82	480	3.3e-16	own
31	61	642	0.82	511	5.6e-17	own
32	63	692	0.82	544	1.1e-16	own
33	65	732	0.82	577	1.1e-16	own
34	67	780	0.82	612	5.6e-17	own
35	69	826	0.82	647	1.1e-16	own
36	71	872	0.82	684	5.6e-17	own
37	73	922	0.82	721	1.1e-16	own
38	75	980	0.82	760	2.2e-16	own
39	77	1032	0.82	799	5.6e-17	derived: Diallo and Worku 2026, elimination started from their rule dwd2p77_n1049 for the same cell

d = 3

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	6	0.91	6	5.6e-17	own
3	5	13	0.78	13	5.6e-17	own
4	7	26	0.74	26	5.6e-17	own
5	9	48	0.73	43	5.6e-17	own
6	11	82	0.72	68	5.6e-17	own
7	13	128	0.72	99	1.1e-16	own
8	15	188	0.72	140	2.8e-17	own
9	17	266	0.71	189	5.6e-17	own
10	19	360	0.71	250	5.6e-17	own
11	21	476	0.71	321	5.6e-17	own
12	23	612	0.71	406	1.1e-16	own
13	25	776	0.71	503	2.2e-16	own
14	27	964	0.71	616	5.6e-17	derived: Diallo and Worku 2026, elimination started from their rule dwd3p27_n984 for the same cell
15	29	1184	0.71	743	1.1e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p29_n1258 for the same cell
16	31	1432	0.70	888	2.2e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p31_n1478 for the same cell
17	33	1714	0.70	1049	2.2e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p33_n1787 for the same cell
18	35	2028	0.70	1230	3.3e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p35_n2102 for the same cell
19	37	2380	0.70	1429	4.4e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p37_n2506 for the same cell
20	39	2770	0.70	1650	2.2e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p39_n2856 for the same cell
21	41	3200	0.70	1891	2.2e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p41_n3338 for the same cell
22	43	3704	0.70	2156	2.2e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p43_n3870 for the same cell
23	45	4308	0.71	2443	1.1e-16	derived: Diallo and Worku 2026, elimination started from their rule dwd3p45_n4414 for the same cell

d = 4

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	8	0.84	8	5.6e-17	own
3	5	21	0.71	21	5.6e-17	own
4	7	54	0.68	48	5.6e-17	own
5	9	120	0.66	91	2.2e-16	own
6	11	234	0.65	160	1.1e-16	own
7	13	416	0.65	259	5.6e-17	own
8	15	690	0.64	400	1.1e-16	own
9	17	1078	0.64	589	2.2e-16	own
10	19	1612	0.63	840	1.1e-16	own
11	21	2322	0.63	1161	1.1e-16	own
12	23	3244	0.63	1568	1.1e-16	own

d = 5

q	p	n	ρ	Möller	rel. err.	origin
1	1	1	1.00	1	0.0e+00	own
2	3	10	0.79	10	1.1e-16	own
3	5	32	0.67	31	5.6e-17	own
4	7	100	0.63	80	5.6e-17	own
5	9	266	0.61	171	2.2e-16	own
6	11	602	0.60	332	5.6e-17	own
7	13	1212	0.59	591	1.1e-16	own
8	15	2500	0.60	992	5.6e-17	own
9	17	3872	0.58	1581	2.2e-16	own
10	19	6826	0.58	2422	1.1e-16	own
11	21	10984	0.58	3583	3.3e-16	own