Keywords: A(n,d), binary codes, bounds, clique-finding, nonlinear codes, perfect codes

A(n,d) is the size of the largest binary code of length n and minimal distance d (the most fundamental quantity in coding theory).
This file contains a table of lower bounds (and in some cases the exact values) of A(n,d).
We are also planning to include explicit codes for as many of these entries as we can locate.

Last updated November 24, 1999.

Simon Litsyn, Tel-Aviv University
Department of Electrical Engineering-Systems, Ramat-Aviv 69978, Tel-Aviv, Israel
Email address: [email protected]; Home page

E. M. Rains, AT&T Labs-Research
Room C290, 180 Park Ave., PO Box 971, Florham Park NJ 07932-0971 USA
Email address: [email protected]; Home page

and

N. J. A. Sloane, AT&T Labs-Research
Room C233, 180 Park Ave., PO Box 971, Florham Park NJ 07932-0971 USA
Email address: [email protected]; Home page

Notes

1. Contributions of new records, explicit codes or upper bounds are welcomed. All contributions will be acknowledged.

2. Format of Table

   Format of table is
   n       d       k       N       Type       Reference meaning that
   A(n,d) >= N*2^k. If n is missing, let the next entry be
   n0       d       k       N       Type       Reference then
   A(n,d) >= N*2^(k-(n0-n)).

3. The table covers the range minimal distance d <= 29 and length n <= 512.

4. The primary source for this table is Simon Litsyn's chapter "Tables of Best Known Binary Codes" which will soon be published in the Handbook of Coding Theory (edited V. Pless et al., North-Holland, 1998, to appear). That table was in turn based on earlier tables of F. J. MacWilliams and N. J. A. Sloane (see for example Appendix A of The Theory of Error-Correcting Codes, North-Holland, 1977).

5. In the earlier tables the codes were described by giving the redundancy r, rather than the number of codewords M = N*2^k. These quantities are related by
   r   =   n   -   log_2(M)   =   n   -   k   -   log_2(N).

   At the end of this page we give a table to assist in converting between the values of r and M.

6. The table only considers odd values of d, since (if d is odd) A(n, d) = A(n+1, d+1).

7. There is a companion table of A(n,d,w), the size of the largest binary code of length n, minimal distance d and constant weight w.

8. See also our home pages: SL | EMR | NJAS

Memo to algorithms specialists:

This file contains a large number of lower bounds on this clique size. If you can improve any of these entries or establish the optimality of any entries that we don't already know are optimal (these are indicated by a period after the number) please let us know (send us the clique too!).

TABLE d=3 Lower bounds on A(n,3)

n	d	k	N	Type	Reference
7	3	4	1	HG	[hamm50]
9	3	3	5	SW	[best80]
11	3	4	9	SW	[juli65]
15	3	11	1	HG	[hamm50]
16	3	5	85	SW	[roma83]
17	3	6	83	SW	[etzi00]
18	3	8	41	MT	[hama88,kaba88]
19	3	12	5	UV	[best80]
23	3	15	9	UV	[sloa70a]
31	3	26	1	HG	[hamm50]
33	3	21	85	UV	[roma83]
35	3	23	83	UV	[etzi00]
37	3	26	41	UV	[hama88,kaba88]
39	3	31	5	UV	[best80]
47	3	38	9	UV	[sloa70a]
63	3	57	1	HG	[hamm50]
70	3	43	1657009	MT	[kaba88]
71	3	63	3	MT	[oste96]
75	3	63	41	UV	[hama88,kaba88]
79	3	70	5	UV	[best80]
95	3	85	9	UV	[sloa70a]
127	3	120	1	HG	[hamm50]
141	3	113	1657009	UV	[kaba88]
143	3	134	3	UV	[oste96]
151	3	138	41	UV	[hama88,kaba88]
159	3	149	5	UV	[best80]
163	3	151	19	MT	[oste96]
191	3	180	9	UV	[sloa70a]
255	3	247	1	HG	[hamm50]
270	3	202	1021273028302258913	MT	[kaba88]
283	3	254	1657009	UV	[kaba88]
287	3	277	3	UV	[oste96]
300	3	220	3348824985082075276195	MT	[kaba88]
303	3	289	41	UV	[hama88,kaba88]
319	3	308	5	UV	[best80]
327	3	314	19	UV	[oste96]
383	3	371	9	UV	[sloa70a]
511	3	502	1	HG	[hamm50]
512	3	443	1021273028302258913	MT	[kaba88]
n	d	k	N	Type	Reference

TABLE d=5 Lower bounds on A(n,5)

n	d	k	N	Type	Reference
8	5	2	1	LI	[cord67]
11	5	3	3	HD	[plot60]
15	5	8	1	PR	[nord67]
19	5	11	1	XP	[sema73,sloa72a]
20	5	9	5	XP	[sloa72a]
23	5	14	1	LI	[wagn66]
33	5	23	1	LI	[chen91]
35	5	22	5	YP	[krac83a]
36	5	20	33	YG	[zino85]
37	5	19	129	YG	[zino85]
63	5	52	1	PR	[prep68]
70	5	58	1	XP	[sema73,sloa72a]
81	5	68	1	LI	[shea]
128	5	114	1	B	[sloa72a]
135	5	117	9	YP	[krac83a]
136	5	114	133	YG	[zino85]
137	5	112	1057	YG	[zino85]
138	5	114	525	YG	[zino85]
139	5	114	1043	YG	[zino85]
140	5	117	259	YG	[zino85]
141	5	116	1029	YG	[zino85]
255	5	240	1	PR	[prep68]
271	5	255	1	XP	[sema73,sloa72a]
278	5	261	1	GB	[chie75]
512	5	494	1	B	[sloa72a]
n	d	k	N	Type	Reference

TABLE d=7 Lower bounds on A(n,7)

n	d	k	N	Type	Reference
11	7	2	1	LI	[cord67]
15	7	5	1	RM	[mull54]
16	7	2	9	CM	[sloa70a]
23	7	12	1	GO	[gola49]
24	7	12	1	-	-
27	7	14	1	DC	[karl69]
31	7	17	1	CS	[chen89]
37	7	22	1	LI	[shea]
39	7	21	5	YG	[krac83a]
63	7	47	1	IM	[goet74]
69	7	52	1	XG	[krac83b]
70	7	49	11	XG	[zino82]
75	7	57	1	XH	[krac83b]
76	7	54	11	XH	[zino82]
88	7	69	1	LI	[shea]
95	7	75	1	LI	[shea]
128	7	107	1	GP	[gopp70]
143	7	118	9	YG	[krac83a]
255	7	233	1	IM	[goet74]
264	7	241	1	XG	[kats87]
273	7	249	1	XH	[kats87]
274	7	249	1	-	-
311	7	285	1	SV	[helg72]
512	7	485	1	GP	[gopp70]
n	d	k	N	Type	Reference

TABLE d=9 Lower bounds on A(n,9)

n	d	k	N	Type	Reference
14	9	2	1	LI	[cord67]
15	9	2	1	-	-
16	9	1	3	HD	[leve61]
19	9	3	5	HD	[leve61]
20	9	1	21	NL	[kaik89]
21	9	1	25	NL	[leve61]
22	9	2	19	NL	[kaik97]
23	9	7	1	LI	[hash76]
24	9	4	11	NL	[kaik97]
25	9	1	135	NL	[kaik97]
27	9	10	1	CY	[pire80]
28	9	2	365	NL	[kaik97]
31	9	13	1	LI	[shea]
35	9	15	3	YC	[sloa72a]
36	9	15	3	-	-
37	9	14	11	Y1	[zino84a]
38	9	16	5	Y1	[zino84a]
39	9	16	9	Y1	[zino84a]
41	9	21	1	QR	[berl68]
42	9	20	3	Y1	[zino84a]
43	9	20	5	Y1	[zino84a]
45	9	24	1	QR	[karl69]
46	9	21	9	Y1	[lits98]
47	9	21	17	Y1	[lits98]
48	9	26	1	LI	[shea]
49	9	23	15	YC	[sloa72a]
50	9	25	7	Y1	[lits98]
51	9	25	13	Y1	[lits98]
52	9	28	3	Y1	[lits98]
53	9	27	11	Y1	[lits98]
54	9	29	5	Y1	[lits98]
55	9	29	9	Y1	[lits98]
56	9	33	1	Y1	[lits98]
57	9	31	7	Y1	[lits98]
58	9	33	3	Y1	[lits98]
59	9	33	5	Y1	[lits98]
64	9	40	1	GP	[macw77]
71	9	46	1	XB	[bier97]
76	9	50	1	XQ	[kasa75]
77	9	47	9	XQ	[zino84b]
91	9	64	1	AL	[helg74]
128	9	100	1	B	[sloa72a]
135	9	106	1	XB	[sloa72b]
136	9	100	93	XB	[zino82]
142	9	112	1	XQ	[kasa75]
143	9	109	9	XQ	[zino84b]
167	9	136	1	AL	[helg74]
256	9	224	1	B	[sloa72a]
265	9	232	1	GP	[sugi76]
275	9	241	1	XB	[zino82]
315	9	280	1	YB	[cace91,rodi]
512	9	476	1	B	[sloa72a]
n	d	k	N	Type	Reference

TABLE d=11 Lower bounds on A(n,11)

n	d	k	N	Type	Reference
17	11	2	1	LI	[cord67]
18	11	2	1	-	-
19	11	1	3	HD	[leve61]
20	11	3	1	B	[gold68]
23	11	4	3	HD	[leve61]
24	11	2	13	CM	[sloa70b]
26	11	7	1	HS	[helg73a]
27	11	1	89	NL	[kaik97]
31	11	11	1	B	[bose60,hoqu59]
33	11	12	1	LI	[shea]
36	11	14	1	LI	[mori93]
47	11	24	1	QR	[karl69]
49	11	24	1	-	-
63	11	37	1	Z4	[cald94,cald96]
64	11	37	1	-	-
67	11	39	1	XB	[sloa72b]
68	11	39	1	-	-
69	11	38	3	XB	[zino82]
71	11	41	1	ZV	[zino76]
74	11	43	1	XB	[tolh86]
75	11	42	3	XB	[zino82]
78	11	46	1	XB	[zino82]
89	11	56	1	CY	[prom78]
99	11	65	1	GP	[roel82]
128	11	93	1	GP	[gopp70]
135	11	99	1	XB	[sloa72b]
136	11	96	9	XB	[zino82]
142	11	105	1	XQ	[kasa75]
143	11	99	93	XQ	[zino84b]
149	11	111	1	GP	[kasa75]
150	11	108	9	GP	[zino84b]
191	11	152	1	YB	[helg73b]
256	11	216	1	GP	[gopp70]
264	11	223	1	XB	[sloa72b]
265	11	216	151	XB	[zino82]
272	11	230	1	XQ	[kasa75]
273	11	223	151	XQ	[zino84b]
280	11	237	1	GP	[sugi76]
281	11	230	151	GP	[zino84b]
311	11	267	1	SV	[helg72]
512	11	467	1	GP	[gopp70]
n	d	k	N	Type	Reference

TABLE d=13 Lower bounds on A(n,13)

n	d	k	N	Type	Reference
20	13	2	1	LI	[cord67]
21	13	2	1	-	-
22	13	2	1	-	-
23	13	1	3	HD	[leve61]
24	13	3	1	HD	[leve61]
27	13	3	7	HD	[plot60]
28	13	2	15	CM	[sloa70b]
29	13	6	1	RM	[mull54]
32	13	8	1	LI	[hell89]
34	13	9	1	CY	[pire74]
38	13	12	1	LI	[bier95]
39	13	12	1	-	-
43	13	15	1	CY	[berl68]
45	13	16	1	XC	[doug91]
47	13	17	1	CY	[jens85]
63	13	32	1	Z4	[ples95,cald96]
64	13	32	1	-	-
65	13	32	1	-	-
66	13	30	7	Y1	[zino84a]
67	13	32	3	Y1	[zino84a]
68	13	32	5	Y1	[zino84a]
70	13	36	1	XB	[sloa72b]
73	13	38	1	XQ	[kasa75]
74	13	38	1	-	-
75	13	37	3	XQ	[zino84b]
77	13	40	1	QR	[karl69]
79	13	41	1	ZV	[kasa76]
81	13	41	3	XZ	[zino84b]
90	13	51	1	CY	[prom78]
94	13	54	1	LI	[brou93]
101	13	60	1	GP	[roel82]
128	13	86	1	GP	[gopp70]
135	13	92	1	XB	[sloa72b]
136	13	89	9	XB	[zino82]
142	13	98	1	XQ	[kasa75]
143	13	95	9	XQ	[zino84b]
149	13	104	1	GP	[sugi76]
150	13	101	9	GP	[zino84b]
156	13	110	1	GP	[sugi76]
157	13	107	9	GP	[zino84b]
191	13	144	1	YB	[helg73b]
256	13	208	1	B	[sloa72a]
264	13	215	1	XB	[sloa72b]
265	13	208	151	XB	[zino82]
272	13	222	1	XQ	[kasa75]
273	13	215	151	XQ	[zino84b]
280	13	229	1	GP	[sugi76]
281	13	222	151	GP	[zino84b]
288	13	236	1	GP	[sugi76]
289	13	229	151	GP	[zino84b]
327	13	274	1	YB	[helg73b]
512	13	458	1	B	[dels75]
n	d	k	N	Type	Reference

TABLE d=15 Lower bounds on A(n,15)

n	d	k	N	Type	Reference
23	15	2	1	LI	[cord67]
25	15	2	1	-	-
26	15	1	3	HD	[leve61]
27	15	3	1	HD	[leve61]
28	15	1	5	HD	[leve61]
31	15	6	1	RM	[mull54]
32	15	6	1	-	-
33	15	3	9	HD	[leve61]
35	15	8	1	LI	[dodu87]
37	15	9	1	LI	[fark94]
39	15	10	1	ZV	[sugi76,zino76]
40	15	8	5	XB	[zino82]
44	15	14	1	LI	[bouk95]
45	15	14	1	-	-
48	15	16	1	LI	[fark94]
50	15	17	1	CY	[berl68]
55	15	21	1	CY	[berl68]
63	15	28	1	CY	[kasa69]
72	15	36	1	CY	[prom78]
73	15	36	1	-	-
74	15	36	1	-	-
79	15	40	1	QR	[karl69]
80	15	40	1	-	-
81	15	37	9	Y1	[zino84a]
82	15	41	1	Y1	[zino84a]
83	15	39	7	Y1	[zino84a]
84	15	41	3	Y1	[zino84a]
85	15	41	5	Y1	[zino84a]
87	15	45	1	QR	[karl69]
88	15	45	1	-	-
89	15	45	1	-	-
90	15	45	1	-	-
91	15	42	11	Y1	[zino84a]
92	15	44	5	Y1	[zino84a]
93	15	44	9	Y1	[zino84a]
94	15	48	1	Y1	[zino84a]
95	15	40	497	YB	[zino85]
96	15	48	3	Y1	[zino84a]
97	15	48	5	Y1	[zino84a]
99	15	52	1	QR	[karl69]
100	15	44	379	Y2	[zino84a]
101	15	50	11	Y2	[zino84a]
102	15	47	163	Y2	[zino84a]
103	15	47	301	Y2	[zino84a]
104	15	48	277	Y2	[zino84a]
105	15	57	1	GP	[roel82]
106	15	53	29	Y2	[zino84a]
107	15	51	211	Y2	[zino84a]
108	15	52	191	Y2	[zino84a]
109	15	55	43	Y2	[zino84a]
110	15	55	77	Y2	[zino84a]
111	15	55	137	Y2	[zino84a]
128	15	79	1	GP	[gopp70]
135	15	85	1	XB	[sloa72b]
136	15	82	9	XB	[zino82]
142	15	91	1	XQ	[kasa75]
143	15	88	9	XQ	[zino84b]
149	15	97	1	GP	[sugi76]
150	15	94	9	GP	[zino84b]
156	15	103	1	GP	[sugi76]
157	15	100	9	GP	[zino84b]
163	15	109	1	GP	[sugi76]
164	15	106	9	GP	[zino84b]
191	15	136	1	YB	[helg73b]
256	15	200	1	GP	[gopp70]
264	15	207	1	XB	[sloa72b]
265	15	200	151	XB	[zino82]
272	15	214	1	XQ	[kasa75]
273	15	207	151	XQ	[zino84b]
280	15	221	1	GP	[sugi76]
281	15	214	151	GP	[zino84b]
288	15	228	1	GP	[sugi76]
289	15	221	151	GP	[zino84b]
296	15	235	1	GP	[sugi76]
297	15	228	151	GP	[zino84b]
327	15	265	1	YB	[helg73b]
512	15	449	1	GP	[gopp70]
n	d	k	N	Type	Reference

TABLE d=17 Lower bounds on A(n,17)

n	d	k	N	Type	Reference
26	17	2	1	LI	[cord67]
27	17	2	1	-	-
28	17	2	1	-	-
29	17	1	3	HD	[leve61]
30	17	1	3	-	-
31	17	3	1	HD	[leve61]
32	17	2	3	-	-
35	17	3	9	HD	[plot60]
36	17	2	19	CM	[sloa70b]
37	17	4	5	HD	[leve61]
38	17	2	21	CM	[sloa70b]
39	17	7	1	LI	[tilb81]
41	17	8	1	GP	[allt77]
44	17	10	1	CY	[gull94]
46	17	11	1	YQ	[gron94]
50	17	12	5	YQ	[lits98]
51	17	12	5	-	-
53	17	16	1	GP	[loel84]
54	17	16	1	-	-
56	17	17	1	LI	[mori94]
60	17	20	1	CY	[weij97]
65	17	24	1	LI	[gron92]
66	17	24	1	-	-
67	17	23	3	Y1	[zino84a]
68	17	23	5	Y1	[zino84a]
71	17	28	1	YC	[sloa72a]
89	17	45	1	QR	[karl69]
90	17	45	1	-	-
91	17	45	1	-	-
92	17	45	1	-	-
93	17	42	11	Y1	[zino84a]
94	17	44	5	Y1	[zino84a]
95	17	44	9	Y1	[zino84a]
96	17	48	1	Y1	[zino84a]
97	17	46	7	Y1	[zino84a]
98	17	48	3	Y1	[zino84a]
99	17	48	5	Y1	[zino84a]
101	17	52	1	QR	[karl69]
102	17	52	1	-	-
103	17	50	5	Y1	[zino84a]
105	17	54	1	DC	[karl69]
106	17	51	11	Y1	[zino84a]
107	17	51	21	Y1	[zino84a]
108	17	54	5	Y1	[zino84a]
109	17	53	19	Y1	[zino84a]
110	17	55	9	Y1	[zino84a]
111	17	55	17	Y1	[zino84a]
112	17	60	1	Y1	[zino84a]
113	17	57	15	Y1	[zino84a]
114	17	59	7	Y1	[zino84a]
115	17	59	13	Y1	[zino84a]
116	17	62	3	Y1	[zino84a]
117	17	61	11	Y1	[zino84a]
118	17	63	5	Y1	[zino84a]
119	17	63	9	Y1	[zino84a]
120	17	67	1	Y1	[zino84a]
121	17	65	7	Y1	[zino84a]
122	17	67	3	Y1	[zino84a]
123	17	67	5	Y1	[zino84a]
125	17	71	1	B	[bose60,hoqu59]
126	17	71	1	-	-
128	17	72	1	B	[sloa72a]
135	17	78	1	XB	[sloa72b]
136	17	75	9	XB	[zino82]
142	17	84	1	XQ	[kasa75]
143	17	81	9	XQ	[zino84b]
149	17	90	1	GP	[sugi76]
150	17	87	9	GP	[zino84b]
156	17	96	1	GP	[sugi76]
157	17	93	9	GP	[zino84b]
163	17	102	1	GP	[sugi76]
164	17	99	9	GP	[zino84b]
170	17	108	1	GP	[sugi76]
171	17	105	9	GP	[zino84b]
191	17	128	1	YB	[helg73b]
256	17	192	1	B	[sloa72a]
264	17	199	1	XB	[sloa72b]
265	17	192	151	XB	[zino82]
272	17	206	1	XQ	[kasa75]
273	17	199	151	XQ	[zino84b]
280	17	213	1	GP	[sugi76]
281	17	206	151	GP	[zino84b]
288	17	220	1	GP	[sugi76]
289	17	213	151	GP	[zino84b]
296	17	227	1	GP	[sugi76]
297	17	220	151	GP	[zino84b]
304	17	234	1	GP	[sugi76]
305	17	227	151	GP	[zino84b]
331	17	260	1	YB	[helg73b]
512	17	440	1	B	[sloa72a]
n	d	k	N	Type	Reference

TABLE d=19 Lower bounds on A(n,19)

n	d	k	N	Type	Reference
29	19	2	1	LI	[cord67]
32	19	3	1	-	-
33	19	1	3	HD	[leve61]
34	19	3	1	CY	[chen70]
35	19	1	5	HD	[leve61]
36	19	2	3	HD	[leve61]
39	19	4	5	HD	[plot60]
40	19	2	21	CM	[sloa70b]
41	19	3	11	HD	[leve61]
42	19	7	1	LI	[tilb81]
44	19	8	1	AX	[allt77]
47	19	10	1	CY	[gull97]
49	19	11	1	YQ	[gron94]
55	19	16	1	GP	[loel84]
56	19	16	1	-	-
57	19	16	1	-	-
59	19	17	1	DG	[dels75]
61	19	18	1	B	[bose60,hoqu59]
65	19	21	1	LI	[zhi93]
72	19	27	1	CY	[prom78]
73	19	27	1	-	-
75	19	28	1	LI	[brou93a]
76	19	28	1	-	-
80	19	31	1	YQ	[gron94]
86	19	36	1	YQ	[gron94]
103	19	52	1	QR	[karl69]
104	19	52	1	-	-
107	19	54	1	QR	[karl69]
108	19	54	1	-	-
109	19	54	1	-	-
127	19	71	1	B	[bose60,hoqu59]
128	19	71	1	-	-
129	19	71	1	-	-
131	19	72	1	GP	[sugi76]
135	19	75	1	GP	[sugi76]
137	19	74	5	XB	[zino82]
138	19	74	9	XB	[zino82]
139	19	78	1	XC	[sloa72a]
140	19	78	1	-	-
141	19	72	69	XB	[kats87]
142	19	79	1	XB	[bier97]
144	19	80	1	XB	[bier97]
145	19	77	9	XB	[zino82]
147	19	82	1	NL	[sugi76]
148	19	80	5	XB	[zino82]
159	19	93	1	XB	[zino84a]
191	19	124	1	YB	[helg73b]
255	19	187	1	B	[bose60,hoqu59]
260	19	191	1	XB	[sloa72b]
261	19	191	1	-	-
262	19	190	3	XB	[zino82]
263	19	192	1	XB	[zino82]
264	19	189	11	XB	[kats87]
267	19	192	9	XB	[zino82]
271	19	199	1	XB	[zino82]
273	19	199	1	-	-
280	19	205	1	GP	[sugi76]
281	19	198	151	GP	[zino84b]
288	19	212	1	GP	[sugi76]
289	19	205	151	GP	[zino84b]
296	19	219	1	GP	[sugi76]
297	19	212	151	GP	[zino84b]
304	19	226	1	GP	[sugi76]
305	19	219	151	GP	[zino84b]
319	19	240	1	GP	[zino84a]
383	19	303	1	GP	[zino84a]
512	19	431	1	GP	[gopp70]
n	d	k	N	Type	Reference

TABLE d=21 Lower bounds on A(n,21)

n	d	k	N	Type	Reference
32	21	2	1	LI	[cord67]
35	21	3	1	-	-
36	21	1	3	HD	[leve61]
37	21	1	3	-	-
38	21	3	1	HD	[leve61]
39	21	1	5	HD	[leve61]
40	21	1	7	HD	[leve61]
43	21	3	11	HD	[plot60]
44	21	3	11	-	-
45	21	5	3	HD	[leve61]
47	21	8	1	GP	[allt76]
48	21	8	1	-	-
49	21	8	1	-	-
50	21	5	13	YD	[zino85]
51	21	8	3	YD	[zino85]
52	21	7	11	YD	[zino85]
53	21	9	5	YD	[zino85]
54	21	9	9	YD	[zino85]
55	21	13	1	YD	[zino85]
56	21	10	11	YD	[zino85]
57	21	9	43	YD	[zino85]
58	21	13	3	Y1	[zino84a]
59	21	13	5	Y1	[zino84a]
61	21	17	1	YD	[dels75]
63	21	18	1	B	[bose60,hoqu59]
64	21	18	1	-	-
65	21	18	1	-	-
66	21	18	1	-	-
68	21	19	1	ZV	[wise83]
71	21	21	1	ZV	[wise83]
74	21	23	1	ZV	[wise83]
77	21	25	1	ZV	[wise83]
80	21	27	1	ZV	[wise83]
83	21	29	1	CY	[jens85]
86	21	31	1	B	[hart73]
87	21	31	1	-	-
88	21	31	1	-	-
90	21	32	1	LI	[kupp95]
92	21	33	1	CY	[prom78]
95	21	35	1	ZV	[zino76]
100	21	39	1	XZ	[chen87]
101	21	36	11	ZV	[zino76]
105	21	43	1	YB	[helg73b]
127	21	64	1	B	[bose60,hoqu59]
135	21	71	1	XB	[sloa72b]
136	21	71	1	-	-
137	21	65	69	XB	[zino82]
138	21	67	23	XB	[zino82]
139	21	72	1	XB	[zino82]
142	21	74	1	GP	[sugi76]
144	21	73	5	XB	[zino82]
145	21	73	9	XB	[zino82]
146	21	73	9	-	-
148	21	73	19	XB	[zino82]
149	21	78	1	XB	[zino82]
150	21	78	1	-	-
151	21	76	5	GP	[zino84b]
152	21	76	9	GP	[zino84b]
154	21	81	1	NL	[sugi76]
155	21	79	5	GP	[zino84b]
158	21	84	1	XQ	[kasa75]
191	21	116	1	YB	[helg73b]
255	21	179	1	B	[bose60,hoqu59]
264	21	187	1	XB	[sloa72b]
265	21	180	151	XB	[zino82]
268	21	190	1	XQ	[kasa75]
269	21	190	1	-	-
270	21	189	3	GP	[zino84b]
271	21	191	1	XQ	[krac83b]
272	21	188	11	XB	[kats87]
275	21	191	9	GP	[zino84b]
279	21	198	1	GP	[zino84b]
280	21	198	1	-	-
281	21	198	1	-	-
288	21	204	1	GP	[sugi76]
289	21	197	151	GP	[zino84b]
296	21	211	1	GP	[sugi76]
297	21	204	151	GP	[zino84b]
304	21	218	1	GP	[sugi76]
305	21	211	151	GP	[zino84b]
312	21	225	1	GP	[sugi76]
313	21	218	151	GP	[zino84b]
320	21	232	1	GP	[sugi76]
321	21	225	151	GP	[zino84b]
383	21	294	1	GP	[zino84a]
512	21	422	1	B	[sloa72a]
n	d	k	N	Type	Reference

TABLE d=23 Lower bounds on A(n,23)

n	d	k	N	Type	Reference
35	23	2	1	LI	[cord67]
36	23	2	1	-	-
37	23	2	1	-	-
38	23	2	1	-	-
39	23	1	3	HD	[leve61]
40	23	1	3	-	-
41	23	3	1	HD	[leve61]
42	23	3	1	-	-
43	23	2	3	HD	[leve61]
44	23	4	1	CY	[chen70]
47	23	5	3	HD	[leve61]
48	23	2	25	CM	[sloa70b]
50	23	8	1	CY	[chen70]
51	23	8	1	-	-
52	23	8	1	-	-
54	23	8	3	YD	[zino85]
55	23	10	1	ZV	[zino76]
57	23	9	7	YD	[zino85]
63	23	17	1	DG	[dels75]
64	23	17	1	-	-
66	23	18	1	XQ	[sloa72b]
69	23	19	1	-	-
71	23	19	1	ZV	[wise83]
74	23	21	1	ZV	[wise83]
77	23	23	1	ZV	[wise83]
80	23	25	1	ZV	[wise83]
83	23	27	1	ZV	[wise83]
87	23	30	1	LI	[karl69]
89	23	31	1	ZV	[wise83]
90	23	31	1	-	-
91	23	31	1	-	-
92	23	31	1	-	-
95	23	33	1	ZV	[zino76]
98	23	35	1	ZV	[tolh86]
101	23	37	1	LI	[brou93]
104	23	39	1	ZV	[brou97]
107	23	41	1	ZV	[chen87]
116	23	49	1	CY	[scho92]
117	23	49	1	-	-
119	23	50	1	ZV	[chen87]
127	23	57	1	B	[bose60,hoqu59]
135	23	64	1	XB	[sloa72b]
136	23	61	9	XB	[zino82]
142	23	70	1	XQ	[kasa75]
143	23	70	1	-	-
144	23	64	69	GP	[zino84b]
145	23	67	15	XB	[zino82]
146	23	71	1	XB	[zino82]
149	23	73	1	GP	[sugi76]
151	23	72	5	GP	[zino84b]
152	23	72	9	GP	[zino84b]
153	23	76	1	GP	[sugi76]
154	23	76	1	-	-
155	23	72	19	GP	[zino84b]
156	23	72	19	-	-
161	23	81	1	ZV	[kasa76]
162	23	77	17	XZ	[zino82]
167	23	86	1	ZV	[kasa76]
168	23	82	17	XZ	[zino82]
173	23	91	1	ZV	[kasa76]
174	23	87	17	XZ	[zino82]
207	23	124	1	YB	[helg73b]
255	23	171	1	B	[bose60,hoqu59]
264	23	179	1	XB	[sloa72b]
265	23	172	151	XB	[zino82]
272	23	186	1	XQ	[kasa75]
273	23	179	151	XZ	[zino84b]
276	23	189	1	GP	[sugi76]
277	23	189	1	-	-
278	23	188	3	GP	[zino84b]
279	23	190	1	GP	[zino84b]
280	23	187	11	XB	[kats87]
283	23	190	9	GP	[zino84b]
287	23	197	1	GP	[zino84b]
288	23	197	1	-	-
289	23	197	1	-	-
296	23	203	1	GP	[sugi76]
297	23	196	151	GP	[zino84b]
304	23	210	1	GP	[sugi76]
305	23	203	151	GP	[zino84b]
312	23	217	1	GP	[sugi76]
313	23	210	151	GP	[zino84b]
320	23	224	1	GP	[sugi76]
321	23	217	151	GP	[zino84b]
328	23	231	1	GP	[sugi76]
329	23	224	151	GP	[zino84b]
383	23	285	1	YB	[zino84a]
512	23	413	1	GP	[gopp70]
n	d	k	N	Type	Reference

TABLE d=25 Lower bounds on A(n,25)

n	d	k	N	Type	Reference
38	25	2	1	LI	[cord67]
39	25	2	1	-	-
40	25	2	1	-	-
41	25	2	1	-	-
42	25	2	1	-	-
43	25	1	3	HD	[leve61]
44	25	1	3	-	-
45	25	3	1	HD	[leve61]
46	25	1	5	HD	[leve61]
47	25	2	3	HD	[leve61]
48	25	4	1	HD	[leve61]
51	25	3	13	HD	[plot60]
52	25	2	27	CM	[sloa70b]
53	25	4	7	HD	[leve61]
55	25	8	1	YK	[zino85]
56	25	3	47	YK	[zino85]
57	25	2	187	YK	[zino85]
58	25	8	3	Y1	[zino84a]
59	25	8	5	Y1	[zino84a]
61	25	12	1	DG	[dels75]
63	25	12	1	-	-
65	25	13	1	CY	[chen70]
67	25	12	5	Y1	[zino84a]
69	25	16	1	YD	[zino82]
70	25	14	5	YD	[zino82]
71	25	17	1	DG	[dels75]
72	25	17	1	-	-
74	25	18	1	XQ	[allt84]
75	25	18	1	-	-
76	25	18	1	-	-
77	25	18	1	-	-
80	25	20	1	CY	[chen94]
82	25	21	1	CY	[chen94]
84	25	22	1	LI	[gron92]
86	25	23	1	YB	[helg73b]
87	25	23	1	-	-
89	25	24	1	LI	[mori94]
91	25	25	1	LI	[gron92]
92	25	25	1	-	-
95	25	27	1	XZ	[barg87]
98	25	29	1	LI	[gron92]
99	25	29	1	-	-
102	25	31	1	LI	[gron92]
111	25	39	1	YB	[zino85]
112	25	39	1	-	-
113	25	36	15	Y1	[zino84a]
114	25	38	7	Y1	[zino84a]
115	25	38	13	Y1	[zino84a]
116	25	41	3	Y1	[zino84a]
117	25	40	11	Y1	[zino84a]
118	25	42	5	Y1	[zino84a]
119	25	42	9	Y1	[zino84a]
120	25	46	1	Y1	[zino84a]
121	25	44	7	Y1	[zino84a]
122	25	46	3	Y1	[zino84a]
123	25	46	5	Y1	[zino84a]
125	25	50	1	B	[bose60,hoqu59]
126	25	50	1	-	-
127	25	50	1	-	-
129	25	1	2047781292273887	XZ	[barg87]
135	25	57	1	XB	[sloa72b]
136	25	54	9	XB	[zino82]
142	25	63	1	XQ	[kasa75]
149	25	69	1	GP	[sugi76]
150	25	69	1	-	-
151	25	63	69	GP	[zino84b]
152	25	66	15	GP	[zino84b]
153	25	70	1	GP	[krac83b]
156	25	72	1	GP	[sugi76]
158	25	71	5	GP	[zino84b]
159	25	71	9	GP	[zino84b]
161	25	76	1	ZV	[kasa76]
162	25	72	17	XZ	[zino82]
167	25	81	1	ZV	[kasa76]
168	25	77	17	XZ	[zino82]
173	25	86	1	ZV	[kasa76]
174	25	82	17	XZ	[zino82]
179	25	91	1	ZV	[kasa76]
180	25	87	17	XZ	[zino82]
185	25	96	1	ZV	[kasa76]
186	25	92	17	XZ	[zino82]
191	25	101	1	ZV	[kasa76]
192	25	97	17	XZ	[zino82]
207	25	116	1	YB	[helg73b]
255	25	163	1	B	[bose60,hoqu59]
264	25	171	1	XB	[sloa72b]
265	25	164	151	XB	[zino82]
272	25	178	1	XQ	[kasa75]
273	25	171	151	XQ	[zino84b]
280	25	185	1	GP	[sugi76]
281	25	178	151	GP	[zino84b]
284	25	188	1	GP	[sugi76]
285	25	188	1	-	-
286	25	187	3	GP	[zino84b]
287	25	189	1	GP	[zino84b]
288	25	186	11	XB	[kats87]
291	25	189	9	GP	[zino84b]
295	25	196	1	GP	[zino84b]
297	25	196	1	-	-
304	25	202	1	GP	[sugi76]
305	25	195	151	GP	[zino84b]
312	25	209	1	GP	[sugi76]
313	25	202	151	GP	[zino84b]
320	25	216	1	GP	[sugi76]
321	25	209	151	GP	[zino84b]
328	25	223	1	GP	[sugi76]
329	25	216	151	GP	[zino84b]
336	25	230	1	GP	[sugi76]
337	25	223	151	GP	[zino84b]
383	25	276	1	YB	[zino84a]
512	25	404	1	B	[sloa72a]
n	d	k	N	Type	Reference

TABLE d=27 Lower bounds on A(n,27)

n	d	k	N	Type	Reference
41	27	2	1	LI	[cord67]
42	27	2	1	-	-
43	27	2	1	-	-
44	27	2	1	-	-
45	27	2	1	-	-
46	27	1	3	HD	[leve61]
47	27	1	3	-	-
48	27	3	1	CY	[chen70]
49	27	3	1	-	-
50	27	1	5	HD	[leve61]
51	27	1	7	HD	[leve61]
52	27	1	9	HD	[leve61]
55	27	4	7	HD	[plot60]
56	27	4	7	-	-
57	27	2	31	YK	[zino85]
63	27	12	1	DG	[dels75]
64	27	12	1	-	-
65	27	12	1	-	-
67	27	13	1	XD	[zino82]
71	27	16	1	XD	[zino82]
72	27	14	5	XD	[zino82]
73	27	17	1	DG	[macw77]
74	27	17	1	-	-
75	27	17	1	-	-
76	27	17	1	-	-
78	27	18	1	XQ	[allt84]
79	27	18	1	-	-
80	27	18	1	-	-
84	27	21	1	CY	[weij97]
85	27	21	1	-	-
88	27	23	1	B	[helg73b]
89	27	23	1	-	-
91	27	24	1	Z4	[aydi99]
92	27	24	1	-	-
93	27	24	1	-	-
95	27	25	1	LI	[gron92]
96	27	25	1	-	-
98	27	26	1	LI	[gron92]
99	27	20	77	Y2	[lits98]
100	27	20	137	Y2	[lits98]
101	27	21	121	Y2	[lits98]
102	27	23	53	Y2	[lits98]
103	27	25	23	Y2	[lits98]
104	27	24	79	Y2	[lits98]
107	27	28	37	YC	[lits98]
108	27	28	37	-	-
109	27	29	29	Y2	[lits98]
110	27	31	11	Y2	[lits98]
112	27	36	1	CY	[scho92]
127	27	50	1	B	[bose60,hoqu59]
128	27	50	1	-	-
129	27	50	1	-	-
131	27	51	1	GP	[sugi76]
135	27	54	1	GP	[sugi76]
137	27	53	5	XB	[zino82]
138	27	53	9	XB	[zino82]
139	27	57	1	XC	[sloa72a]
140	27	57	1	-	-
141	27	51	69	XB	[kats87]
142	27	58	1	XB	[bier97]
144	27	59	1	XB	[bier97]
147	27	61	1	NL	[sugi76]
148	27	59	5	XB	[zino82]
151	27	64	1	GP	[sugi76]
156	27	68	1	GP	[sugi76]
157	27	68	1	-	-
158	27	62	69	GP	[zino84b]
159	27	69	1	XZ	[barg87]
167	27	76	1	ZV	[kasa76]
168	27	72	17	XZ	[zino82]
173	27	81	1	ZV	[kasa76]
174	27	77	17	XZ	[zino82]
179	27	86	1	ZV	[kasa76]
180	27	82	17	XZ	[zino82]
185	27	91	1	ZV	[kasa76]
186	27	87	17	XZ	[zino82]
191	27	96	1	ZV	[kasa76]
192	27	92	17	XZ	[zino82]
197	27	101	1	ZV	[kasa76]
198	27	97	17	XZ	[zino82]
199	27	100	3	XZ	[zino82]
200	27	102	1	XZ	[zino82]
201	27	97	45	XZ	[kats87]
204	27	102	9	XZ	[zino82]
207	27	108	1	YB	[helg73b]
255	27	155	1	B	[bose60,hoqu59]
264	27	163	1	XB	[sloa72b]
265	27	156	151	XB	[zino82]
272	27	170	1	XQ	[kasa75]
273	27	163	151	XQ	[zino84b]
280	27	177	1	GP	[sugi76]
281	27	170	151	GP	[zino84b]
288	27	184	1	GP	[sugi76]
289	27	177	151	GP	[zino84b]
292	27	187	1	GP	[sugi76]
293	27	187	1	-	-
294	27	186	3	GP	[zino84b]
295	27	188	1	GP	[zino84b]
296	27	185	11	XB	[kats87]
299	27	188	9	GP	[zino84b]
303	27	195	1	GP	[zino84b]
304	27	195	1	-	-
305	27	195	1	-	-
312	27	201	1	GP	[sugi76]
313	27	194	151	GP	[zino84b]
320	27	208	1	GP	[sugi76]
321	27	201	151	GP	[zino84b]
328	27	215	1	GP	[sugi76]
329	27	208	151	GP	[zino84b]
336	27	222	1	GP	[sugi76]
337	27	215	151	GP	[zino84b]
344	27	229	1	GP	[sugi76]
345	27	222	151	GP	[zino84b]
383	27	267	1	GP	[sugi76]
512	27	395	1	GP	[gopp70]
n	d	k	N	Type	Reference

TABLE d=29 Lower bounds on A(n,29)

n	d	k	N	Type	Reference
44	29	2	1	LI	[cord67]
45	29	2	1	-	-
46	29	2	1	-	-
47	29	2	1	-	-
48	29	2	1	-	-
49	29	1	3	HD	[leve61]
50	29	1	3	-	-
51	29	1	3	-	-
52	29	3	1	HD	[leve61]
53	29	1	5	HD	[leve61]
54	29	2	3	HD	[leve61]
55	29	1	7	HD	[leve61]
56	29	2	5	HD	[leve61]
59	29	3	15	HD	[plot60]
60	29	2	31	CM	[sloa70b]
61	29	7	1	RM	[mull54]
62	29	7	1	-	-
63	29	3	17	HD	[leve61]
64	29	8	1	LI	[dodu87]
69	29	12	1	XD	[macw77]
70	29	12	1	-	-
71	29	12	1	-	-
73	29	13	1	XD	[zino82]
74	29	13	1	-	-
75	29	11	5	YZ	[zino85]
78	29	16	1	LI	[gron92]
80	29	17	1	XD	[zino82]
81	29	17	1	-	-
82	29	17	1	-	-
83	29	17	1	-	-
85	29	18	1	XB	[zino82]
86	29	18	1	-	-
87	29	18	1	-	-
93	29	23	1	LI	[gron94]
94	29	23	1	-	-
95	29	23	1	-	-
96	29	23	1	-	-
98	29	24	1	LI	[gron94]
100	29	25	1	CY	[weij97]
107	29	28	9	YC	[lits98]
108	29	28	9	-	-
109	29	32	1	Y1	[lits98]
110	29	30	7	Y1	[lits98]
111	29	32	3	Y1	[lits98]
112	29	32	5	Y1	[lits98]
114	29	36	1	CY	[scho92]
115	29	36	1	-	-
117	29	37	1	CY	[scho92]
118	29	35	5	Y1	[zino84a]
119	29	35	9	Y1	[zino84a]
120	29	39	1	Y1	[zino84a]
121	29	37	7	Y1	[zino84a]
122	29	39	3	Y1	[zino84a]
123	29	39	5	Y1	[zino84a]
125	29	43	1	B	[kasa69]
126	29	43	1	-	-
127	29	43	1	-	-
135	29	50	1	XB	[sloa72b]
136	29	50	1	-	-
137	29	44	69	XB	[zino82]
138	29	46	23	XB	[zino82]
139	29	51	1	XB	[zino82]
142	29	53	1	GP	[sugi76]
144	29	52	5	XQ	[zino82]
145	29	52	9	XQ	[zino82]
146	29	56	1	XQ	[kasa75]
147	29	56	1	-	-
148	29	52	19	XB	[zino82]
149	29	57	1	XB	[zino82]
154	29	61	1	XZ	[barg87]
155	29	57	17	XZ	[barg87]
159	29	65	1	XZ	[barg87]
160	29	61	17	XZ	[barg87]
164	29	69	1	XZ	[barg87]
165	29	65	17	XZ	[barg87]
166	29	68	3	XZ	[barg87]
167	29	70	1	ZV	[kasa76]
168	29	67	11	XZ	[kats87]
170	29	67	33	XZ	[barg87]
179	29	81	1	ZV	[kasa76]
180	29	77	17	XZ	[zino82]
185	29	86	1	ZV	[kasa76]
186	29	82	17	XZ	[zino82]
191	29	91	1	ZV	[kasa76]
192	29	87	17	XZ	[zino82]
197	29	96	1	ZV	[kasa76]
198	29	92	17	XZ	[zino82]
199	29	92	29	XZ	[zino84b]
200	29	90	155	Y1	[shek89]
201	29	86	4589	Y1	[shek89]
202	29	87	4237	Y1	[shek89]
203	29	87	7807	Y1	[shek89]
204	29	91	897	Y1	[shek89]
205	29	91	1645	Y1	[shek89]
206	29	91	3009	Y1	[shek89]
207	29	91	5489	Y1	[shek89]
208	29	99	39	Y1	[shek89]
209	29	94	2263	Y1	[shek89]
210	29	95	2045	Y1	[shek89]
211	29	95	3683	Y1	[shek89]
212	29	99	413	Y1	[shek89]
213	29	100	369	Y1	[shek89]
214	29	99	1313	Y1	[shek89]
215	29	99	2325	Y1	[shek89]
216	29	111	1	Y1	[shek89]
217	29	102	897	Y1	[shek89]
218	29	103	781	Y1	[shek89]
219	29	103	1351	Y1	[shek89]
220	29	107	145	Y1	[shek89]
221	29	107	247	Y1	[shek89]
222	29	107	417	Y1	[shek89]
223	29	107	697	Y1	[shek89]
224	29	114	9	Y1	[shek89]
225	29	110	235	Y1	[shek89]
226	29	111	189	Y1	[shek89]
227	29	111	299	Y1	[shek89]
255	29	147	1	B	[bose60,hoqu59]
264	29	155	1	XB	[sloa72b]
265	29	148	151	XB	[zino82]
272	29	162	1	XQ	[kasa75]
273	29	155	151	XQ	[zino84b]
280	29	169	1	GP	[sugi76]
281	29	162	151	GP	[zino84b]
288	29	176	1	GP	[sugi76]
289	29	169	151	GP	[zino84b]
296	29	183	1	GP	[sugi76]
297	29	176	151	GP	[zino84b]
300	29	186	1	GP	[sugi76]
301	29	186	1	-	-
302	29	185	3	GP	[zino84b]
303	29	187	1	GP	[zino84b]
304	29	184	11	XB	[kats87]
307	29	187	9	GP	[zino84b]
311	29	194	1	GP	[zino84b]
313	29	194	1	-	-
320	29	200	1	GP	[sugi76]
321	29	193	151	GP	[zino84b]
328	29	207	1	GP	[sugi76]
329	29	200	151	GP	[zino84b]
336	29	214	1	GP	[sugi76]
337	29	207	151	GP	[zino84b]
344	29	221	1	GP	[sugi76]
345	29	214	151	GP	[zino84b]
352	29	228	1	GP	[sugi76]
353	29	221	151	GP	[zino84b]
383	29	258	1	GP	[zino84a]
512	29	386	1	B	[sloa72a]
n	d	k	N	Type	Reference

KEY

AL = alternant code (linear)
AX = the (a+x, b+x, a+b+x) construction
B = BCH code
BV = Belov et al.'s linear codes meeting the Griesmer bound
CM = conference-matrix code
CS = code from a group
CY = cyclic or quasicyclic code
DC = double circulant code
DG = Delsarte-Goethals code
GB = generalized BCH code
GO = Golay code
GP = Goppa or lengthened Goppa code
HD = Hadamard matrix code
HG = Hamming code
HS = Helgert and Stinaff construction A (Griesmer shortening)
IM = Goethals nonlinear code I(M)
KS = Kasahara et al. generalized concatenated codes (linear or nonlinear)
LI = linear code (a generating or parity-check matrix is explicitly found)
MT = matrix construction
NL = nonlinear code (the list of codewords is explicitly given)
PR = Preparata code
PT = Piret's construction
QR = quadratic residue code
RM = Reed-Muller code
SV = Srivastava code
SW = nonlinear single error-correcting code
UV = the ({\bf u},{\bf u}+{\bf v})-construction
X = lengthening construction X
XB = lengthening of BCH codes
XC = lengthening of a cyclic code
XD = lengthening of Delsarte-Goethals codes
XG = lengthening of Preparata-Goethals codes
XH = lengthening of Hamming-Preparata-Goethals codes
XP = lengthening of Hamming-Preparata codes
XQ = two-dimensional lengthening of BCH codes
XZ = lengthening of generalized concatenated codes
X3 = construction X3
X4 = construction X4
Y1 = shortening of codes with j=1
Y2 = shortening of codes with j=2
YB = shortening of BCH codes
YC = shortening of cyclic or quadratic residue codes
YD = shortening of Delsarte-Goethals codes
YG = shortening of Goethals codes
YK = shortening of Kerdock codes
YP = shortening of Preparata codes
YQ = shortening of Goppa codes
YZ = shortening of generalized concatenated codes
ZV = generalized concatenated codes
Z4 = codes derived from Z_4 cyclic codes.

Table of size, rate, redundancy.

To illustrate the use of this table, consider the second line in the d=3 table, which describes a code of length n = 9, minimal distance d = 3, with k = 3, N = 5, hence has M = 2^3 * 5 = 40 codewords. Referring to the third line (N=5) of the table below, we find that this code has

and

Table of size, rate, redundancy:

N	log_2(N)	red
1	0.00000	1.00000
3	1.58496	0.41504
5	2.32193	0.67807
7	2.80735	0.19265
9	3.16993	0.83007
11	3.45943	0.54057
13	3.70044	0.29956
15	3.90689	0.09311
17	4.08746	0.91254
19	4.24793	0.75207
21	4.39232	0.60768
23	4.52356	0.47644
25	4.64386	0.35614
27	4.75489	0.24511
29	4.85798	0.14202
31	4.95420	0.04580
33	5.04439	0.95561
35	5.12928	0.87072
37	5.20945	0.79055
39	5.28540	0.71460
41	5.35755	0.64245
43	5.42626	0.57374
45	5.49185	0.50815
47	5.55459	0.44541
49	5.61471	0.38529
51	5.67243	0.32757
53	5.72792	0.27208
55	5.78136	0.21864
57	5.83289	0.16711
59	5.88264	0.11736
63	5.97728	0.02272
65	6.02237	0.97763
67	6.06609	0.93391
69	6.10852	0.89148
71	6.14975	0.85025
73	6.18982	0.81018
75	6.22882	0.77118
77	6.26679	0.73321
79	6.30378	0.69622
81	6.33985	0.66015
83	6.37504	0.62496
85	6.40939	0.59061
87	6.44294	0.55706
89	6.47573	0.52427
91	6.50779	0.49221
93	6.53916	0.46084
95	6.56986	0.43014
97	6.59991	0.40009
99	6.62936	0.37064
101	6.65821	0.34179
103	6.68650	0.31350
105	6.71425	0.28575
107	6.74147	0.25853
109	6.76818	0.23182
111	6.79442	0.20558
113	6.82018	0.17982
115	6.84549	0.15451
117	6.87036	0.12964
119	6.89482	0.10518
121	6.91886	0.08114
123	6.94251	0.05749
125	6.96578	0.03422
127	6.98868	0.01132
129	7.01123	0.98877
133	7.05528	0.94472
135	7.07682	0.92318
137	7.09803	0.90197
145	7.17991	0.82009
151	7.23840	0.76160
155	7.27612	0.72388
163	7.34873	0.65127
187	7.54689	0.45311
189	7.56224	0.43776
191	7.57743	0.42257
211	7.72110	0.27890
235	7.87652	0.12348
247	7.94837	0.05163
259	8.01681	0.98319
277	8.11374	0.88626
299	8.22400	0.77600
301	8.23362	0.76638
365	8.51175	0.48825
369	8.52748	0.47252
379	8.56605	0.43395
413	8.69000	0.31000
417	8.70390	0.29610
497	8.95710	0.04290
525	9.03617	0.96383
697	9.44501	0.55499
781	9.60918	0.39082
897	9.80896	0.19104
1029	10.00703	0.99297
1043	10.02652	0.97348
1057	10.04576	0.95424
1313	10.35865	0.64135
1351	10.39981	0.60019
1645	10.68387	0.31613
2045	10.99789	0.00211
2263	11.14402	0.85598
2325	11.18302	0.81698
3009	11.55507	0.44493
3683	11.84667	0.15333
4237	12.04883	0.95117
4589	12.16396	0.83604
5489	12.42233	0.57767
7807	12.93055	0.06945
1657009	20.66015	0.33985
2047781292273887	51.86298	0.13702
1021273028302258913	59.82507	0.17493
3348824985082075276195	71.50414	0.49586