Mathematical Problems in Engineering

Research Article

New Bounds for Ternary Covering Arrays Using a Parallel Simulated Annealing

Table 2

It shows the improved bounds produced by CSA approach. Column represents the best-known solution reported in the literature [30]. Column represents the best solution in terms of produced by CSA approach. Last column () depicts the difference between the best result produced by CSA approach and the best-known solution (). (a) Improved bounds on ; (b) improved bounds on ; (c) improved bounds on ; (d) improved bounds on .

(a)




3	4	27	27	0
3	5	33	33	0
3	6	33	33	0
3	7	40	39	−1
3	8	42	42	0
3	9	45	45	0
3	10	45	45	0
3	11	45	45	0
3	12	45	45	0
3	13	50	49	−1
3	14	51	50	−1
3	15	57	57	0
3	16	60	59	−1
3	17	60	59	−1
3	18	60	59	−1
3	19	60	59	−1
3	20	60	59	−1
3	21	66	67	1
3	22	66	71	5
3	23	69	71	2
3	24	72	71	−1
3	25	75	72	−3
3	26	78	72	−6
3	27	81	79	−2
3	28	81	79	−2
3	29	87	84	−3
3	30	87	84	−3
3	31	90	88	−2
3	32	90	89	−1
3	33	90	89	−1
3	34	90	89	−1
3	35	90	89	−1
3	36	90	89	−1
3	37	90	89	−1
3	38	90	89	−1
3	39	90	89	−1
3	40	90	89	−1
3	41	98	94	−4
3	42	98	94	−4
3	43	100	99	−1
3	44	100	99	−1
3	45	103	99	−4
3	46	103	101	−2
3	47	106	101	−5
3	48	106	101	−5
3	49	108	101	−7
3	50	108	102	−6

(b)




4	5	81	81	0
4	6	111	111	0
4	7	123	123	0
4	8	141	135	−6
4	9	159	135	−24
4	10	159	164	5
4	11	183	183	0
4	12	201	201	0
4	13	219	219	0
4	14	237	249	12
4	15	237	277	40
4	16	237	277	40
4	17	300	287	−13
4	18	307	300	−7
4	19	313	313	0
4	20	315	321	6
4	21	315	338	23
4	22	315	347	32
4	23	315	359	44
4	24	377	375	−2
4	25	384	375	−9
4	26	393	387	−6
4	27	393	387	−6
4	28	393	392	−1
4	29	393	406	13
4	30	393	401	8
4	31	440	424	−16
4	32	445	431	−14
4	33	454	438	−16
4	34	462	447	−15
4	35	471	440	−31
4	36	471	456	−15
4	37	471	460	−11
4	38	471	465	−6
4	39	471	468	−3
4	40	499	472	−27
4	41	506	484	−22
4	42	509	488	−21
4	43	518	494	−24
4	44	522	497	−25
4	45	526	497	−29
4	46	530	506	−24
4	47	534	510	−24
4	48	542	516	−26
4	49	549	523	−26
4	50	549	525	−24

(c)




5	6	243	243	0
5	7	351	351	0
5	8	405	405	0
5	9	483	405	−78
5	10	483	405	−78
5	11	705	550	−155
5	12	723	600	−123
5	13	723	828	105
5	14	922	890	−32
5	15	963	944	−19
5	16	963	1025	62
5	17	1117	1117	0
5	18	1167	1165	−2
5	19	1197	1190	−7
5	20	1266	1257	−9
5	21	1317	1312	−5
5	22	1346	1319	−27
5	23	1405	1387	−18
5	24	1447	1420	−27
5	25	1486	1440	−46
5	26	1521	1493	−28
5	27	1538	1527	−11
5	28	1579	1555	−24
5	29	1615	1585	−30
5	30	1647	1616	−31
5	31	1681	1643	−38
5	32	1724	1671	−53
5	33	1783	1702	−81
5	34	1783	1724	−59
5	35	1851	1748	−103
5	36	1882	1778	−104
5	37	1909	1800	−109
5	38	1937	1829	−108
5	39	1960	1851	−109
5	40	1986	1866	−120
5	41	2023	1896	−127
5	42	2046	1923	−123
5	43	2069	1940	−129
5	44	2091	2089	−2
5	45	2112	2111	−1
5	46	2130	2129	−1
5	47	2150	2149	−1
5	48	2174	2168	−6
5	49	2191	2189	−2
5	50	2213	2211	−2

(d)




6	7	729	729	0
6	8	1152	1152	0
6	9	1431	1600	169
6	10	1449	1849	400
6	11	1449	2136	687
6	12	2181	2482	301
6	13	2734	2744	10
6	14	2907	3220	313
6	15	3234	3338	104
6	16	3443	3672	229
6	17	3658	3882	224
6	18	3846	4098	252
6	19	4054	4256	202
6	20	4486	4400	−86
6	21	4678	4600	−78
6	22	4853	4732	−121
6	23	4942	4941	−1
6	24	5193	5100	−93
6	25	5257	5238	−19
6	26	5709	5380	−329
6	27	5853	5810	−43
6	28	6003	5965	−38
6	29	6150	6110	−40
6	30	6281	6250	−31
6	31	6413	6393	−20
6	32	6535	6518	−17
6	33	6656	6642	−14
6	34	6772	6760	−12
6	35	6877	6871	−6
6	36	6989	6978	−11
6	37	7092	7086	−6
6	38	7194	7187	−7
6	39	7293	7284	−9
6	40	7391	7385	−6
6	41	7490	7478	−12
6	42	7574	7569	−5
6	43	7672	7661	−11
6	44	7757	7748	−9
6	45	7845	7836	−9
6	46	7938	7928	−10
6	47	8013	8005	−8
6	48	8092	8089	−3
6	49	8179	8176	−3
6	50	8256	8253	−3