Abstract
In this paper, we show that for all \(v\equiv 0,1\) (mod 5) and \(v\ge 15\), there exists a super-simple (v, 5, 2) directed design. Moreover, for these parameters, there exists a super-simple (v, 5, 2) directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes.
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Appendix
Appendix
v | Base blocks | \(b_v\) | d | ||||
|---|---|---|---|---|---|---|---|
40 | (7,8,18,5,0) | (0,11,13,27,36) | (0,14,17,2,22) | (0,20,24,30,6) | mod 39 | 312 | \(\frac{4\times 39}{312}\) |
(0,38,22,12,30) | (0,28,\(\infty \), 4,33) | (6,0,1,38,18) | (0,7,26,35,3) | ||||
41 | (0,4,1,11,29) | (6,8,27,0,32) | (0,11,7,10,23) | (0,39,20,6,15) | mod 41 | 328 | \(\frac{4\times 41}{328}\) |
(36,39,28,7,0) | (1,19,0,25,15) | (0,8,38,36,29) | (0,23,1,27,17) | ||||
45 | (0,11,21,2,7) | (0,40,9,26,43) | (0,18,38,32,39) | (0,3,1,29,19) | mod 44 | 396 | \(\frac{4\times 44+22}{396}\) |
(0,4,10,23,12) | (0,41,24,5,36) | (3,9,\(\infty \), 0,32) | (0,16,43,14,36) | ||||
(22,0,9,33,37) | |||||||
46 | (14,1,7,0,10) | (12,0,24,26,32) | (31,0,28,25,29) | (0,5,24,22,45) | mod 46 | 414 | \(\frac{4\times 46+23}{414}\) |
(0,7,2,16,33) | (10,0,15,37,28) | (17,0,25,35,38) | (0,20,1,36,31) | ||||
(23,0,30,42,34) | |||||||
50 | (0,15,8,47,48) | (5,45,0,19,42) | (1,23,0,3,31) | (0,5,25,12,36) | mod 49 | 490 | \(\frac{5\times 49}{490}\) |
(0,32,41,11,45) | (0,30,45,47,16) | (0,38,35,41,10) | (20,0,27,33,21) | ||||
(5,0,17,27,43) | |||||||
(0,5,\(\infty \),14,39) | |||||||
51 | (40,17,0,22,29) | (17,27,30,32,0) | (0,33,20,39,50) | (5,27,35,0,42) | mod 51 | 510 | \(\frac{5\times 51}{510}\) |
(15,0,18,47,31) | (6,42,0,32,44) | (0,4,31,41,8) | (28,2,42,16,0) | ||||
(24,30,0,50,47) | |||||||
(7,8,5,50,0) | |||||||
55 | (15,25,53,3,0) | (6,8,0,28,31) | (14,7,22,0,40) | (6,0,21,42,44) | mod 54 | 594 | \(\frac{5\times 54+27}{594}\) |
(2,20,3,47,0) | (0,9,5,13,16) | (0,20,31,45,30) | (5,11,\(\infty \),0,24) | ||||
(4,13,18,30,0) | (17,0,6,33,52) | ||||||
(5,0,35,22,34) | |||||||
56 | (4,29,0,10,48) | (0,40,21,39,41) | (0,23,15,20,28) | (0,7,29,6,46) | mod 56 | 616 | \(\frac{2\times 56+4\times 56}{616}\) |
(0,9,27,13,30) | (4,2,32,0,36) | (0,51,11,44,26) | (6,20,13,32,0) | ||||
(9,11,25,0,34) | |||||||
(0,10,42,45,53) | |||||||
(0,46,12,47,41) |
v | Base blocks | \(b_v\) | d | ||||
|---|---|---|---|---|---|---|---|
60 | (4,1,0,10,52) | (2,37,5,0,41) | (6,31,0,19,52) | (0,44,\(\infty \),14,23) | mod 59 | 708 | \(\frac{6\times 59}{708}\) |
(3,0,1,17,34) | (0,3,8,21,32) | (0,5,16,42,6) | (16,0,4,40,31) | ||||
(8,0,2,20,45) | (25,10,0,32,12) | ||||||
(0,41,10,40,48) | (5,24,0,20,50) | ||||||
61 | (3,0,55,1,21) | (0,2,6,49,42) | (4,0,12,23,37) | (24,8,0,46,13) | mod 61 | 732 | \(\frac{6\times 61}{732}\) |
(3,0,9,2,43) | (6,0,18,4,25) | (12,0,36,8,50) | (0,16,48,31,26) | ||||
(32,0,35,1,52) | (0,11,24,16,39) | ||||||
(35,0,3,44,34) | (0,48,22,32,17) | ||||||
65 | (10,25,0,23,39) | (23,20,0,22,48) | (0,23,19,53,51) | (0,27,36,35,5) | mod 64 | 832 | \(\frac{6\times 64+32}{832}\) |
(0,41,\(\infty \),31,52) | (0,10,22,27,1) | (0,46,25,19,49) | (0,37,51,31,57) | ||||
(0,7,45,16,56) | (4,49,44,0,56) | (28,14,0,32,61) | |||||
(5,47,7,0,53) | (9,56,0,10,13) | ||||||
66 | (17,37,36,0,65) | (15,27,5,0,24) | (22,4,0,20,27) | (10,0,17,25,43) | mod 66 | 858 | \(\frac{6\times 66+33}{858}\) |
(20,0,31,62,24) | (6,29,63,0,42) | (5,0,35,57,9) | (3,13,0,49,54) | ||||
(34,0,6,32,59) | (45,0,58,39,47) | (32,16,0,33,54) | |||||
(0,3,14,15,55) | (29,0,43,35,45) | ||||||
70 | (0,26,33,52,64) | (5,59,0,25,43) | (18,37,47,0,65) | (30,35,0,37,36) | mod 69 | 966 | \(\frac{7\times 69}{966}\) |
(9,23,38,0,62) | (32,0,44,23,57) | (29,33,0,9,50) | (2,37,\(\infty \),13,0) | ||||
(0,9,11,59,51) | (22,52,0,55,49) | (0,16,13,21,43) | |||||
(4,27,0,20,28) | (6,12,0,4,58) | (1,0,15,56,59) | |||||
71 | (8,48,41,35,0) | (20,0,67,49,52) | (0,66,41,1,58) | (0,40,7,13,48) | mod 71 | 994 | \(\frac{7\times 71}{994}\) |
(0,7,4,27,42) | (47,0,69,19,37) | (1,6,0,31,14) | (23,27,0,20,56) | ||||
(43,0,55,34,45) | (34,0,62,50,60) | (19,0,43,53,21) | |||||
(62,0,16,17,5) | (16,25,0,70,11) | (0,67,15,47,18) |
v | Base blocks | \(b_v\) | d | ||||
|---|---|---|---|---|---|---|---|
75 | (0,30,19,25,47) | (5,1,0,73,57) | (18,36,0,20,47) | (40,55,0,8,60) | mod 74 | 1110 | \(\frac{7\times 74+37}{1110}\) |
(0,44,36,51,57) | (37,0,4,33,55) | (0,36,21,60,61) | (0,23,31,16,28) | ||||
(0,30,39,10,65) | (24,0,72,31,65) | (0,32,1,49,46) | (12,25,28,65,0) | ||||
(32,21,64,0,44) | (0,4,\(\infty \),66,68) | (0,3,14,38,64) | |||||
76 | (36,17,58,0,71) | (5,58,7,0,65) | (0,5,51,55,67) | (15,0,27,42,56) | mod 76 | 1140 | \(\frac{7\times 76+38}{1140}\) |
(0,36,53,20,55) | (0,38,63,37,48) | (6,12,0,5,45) | (27,0,8,37,51) | ||||
(0,9,1,67,22) | (0,3,4,52,23) | (10,0,54,6,52) | (42,0,74,30,68) | ||||
(29,37,52,0,73) | (11,45,0,73,56) | (0,30,17,26,33) | |||||
80 | (21,38,20,0,28) | (32,8,33,0,11) | (0,50,44,51,40) | (0,49,64,4,40) | mod 79 | 1264 | \(\frac{8\times 79}{1264}\) |
(37,70,0,20,68) | (40,0,61,39,63) | (73,0,22,54,20) | (19,0,43,65,13) | ||||
(0,38,\(\infty \),64,10) | (16,4,7,0,52) | (0,2,44,17,76) | (65,23,29,0,33) | ||||
(0,66,41,53,71) | (48,45,61,56,0) | (0,37,53,67,72) | (0,35,62,12,21) | ||||
81 | (49,0,47,26,30) | (27,38,0,65,79) | (7,31,66,68,0) | (0,16,58,21,52) | mod 81 | 1296 | \(\frac{8\times 81}{1296}\) |
(0,41,15,49,31) | (0,50,7,75,78) | (8,0,7,64,18) | (37,42,0,74,46) | ||||
(52,0,17,60,78) | (14,0,20,68,69) | (0,33,45,67,73) | (0,56,40,69,70) | ||||
(9,18,0,45,57) | (28,48,0,51,70) | (57,0,59,80,76) | (9,34,19,4,0) | ||||
85 | (4,7,60,0,82) | (48,0,68,25,11) | (25,41,0,57,58) | (0,18,68,30,47) | mod 84 | 1428 | \(\frac{8\times 84+42}{1428}\) |
(0,10,45,15,75) | (17,35,0,48,73) | (15,0,41,67,74) | (56,38,0,53,42) | ||||
(18,52,57,0,55) | (1,22,0,72,65) | (4,0,78,28,48) | (29,0,60,38,40) | ||||
(5,0,\(\infty \),19,40) | (75,0,76,4,83) | (0,49,72,51,57) | (3,26,48,40,0) | ||||
(0,29,71,39,6) | |||||||
86 | (68,41,72,69,0) | (0,59,84,31,33) | (0,63,43,7,78) | (0,22,77,21,79) | mod 86 | 1462 | \(\frac{8\times 86+43}{1462}\) |
(13,64,0,12,67) | (10,0,72,70,82) | (0,23,41,50,47) | (0,27,5,37,46) | ||||
(30,78,24,0,44) | (0,36,26,78,65) | (79,0,38,74,49) | (17,46,70,0,28) | ||||
(16,0,42,65,19) | (0,8,69,82,75) | (0,39,1,17,54) | (0,20,77,66,71) | ||||
(5,18,39,0,43) | |||||||
95 | (3,77,0,1,36) | (40,1,0,13,4) | (0,80,82,64,69) | (48,0,92,67,59) | mod 94 | 1598 | \(\frac{8\times 94+47}{1598}\) |
(49,53,0,77,84) | (21,73,0,81,38) | (79,0,89,14,10) | (46,7,0,48,75) | ||||
(3,68,86,52,0) | (12,28,62,0,13) | (19,58,0,63,25) | (38,72,1,8,0) | ||||
(0,6,\(\infty \),20,72) | (55,0,26,88,37) | (0,31,88,16,40) | (23,0,3,50,43) | ||||
(32,0,28,23,70) |
v | Base blocks | \(b_v\) | d | ||||
|---|---|---|---|---|---|---|---|
110 | (0,101,30,34,43) | (36,0,103,19,83) | (46,0,74,57,98) | (0,1,44,50,67) | mod 109 | 2398 | \(\frac{11\times 109}{2398}\) |
(0,18,26,40,98) | (40,85,0,88,97) | (40,38,0,31,48) | (4,70,81,14,0) | ||||
(0,100,73,78,85) | (0,33,87,46,53) | (0,15,108,94,75) | (0,59,77,79,81) | ||||
(23,48,63,0,104) | (65,8,0,55,84) | (3,19,\(\infty \),54,0) | (86,37,0,107,102) | ||||
(0,74,71,108,61) | (21,0,86,59,89) | (51,26,0,96,80) | |||||
(0,36,50,85,32) | (0,6,33,97,27) | (0,32,1,37,63) | |||||
111 | (8,0,80,85,104) | (33,69,68,0,76) | (0,55,87,97,57) | (0,33,34,108,86) | mod 111 | 2442 | \(\frac{11\times 111}{2442}\) |
(2,19,50,41,0) | (0,68,5,105,79) | (57,47,0,88,103) | (44,28,104,0,110) | ||||
(0,3,20,107,86) | (51,0,63,91,64) | (23,0,52,96,39) | (79,6,27,77,0) | ||||
(53,36,56,0,80) | (5,0,51,107,41) | (0,59,15,40,77) | (38,83,0,61,65) | ||||
(11,0,93,79,33) | (0,69,81,95,99) | (33,20,82,0,90) | |||||
(0,47,49,58,72) | (17,12,0,38,47) | (0,53,50,63,69) | |||||
115 | (0,106,11,1,109) | (0,113,53,111,102) | (46,16,0,74,1) | (50,43,30,0,42) | mod 114 | 2622 | \(\frac{11\times 114+57}{2622}\) |
(18,50,106,87,0) | (0,4,6,83,29) | (80,98,57,112,0) | (93,100,34,79,0) | ||||
(0,102,12,79,82) | (0,85,11,74,7) | (75,97,68,101,0) | (42,0,62,108,15) | ||||
(51,101,34,0,30) | (88,10,26,0,51) | (54,0,71,18,5) | (36,0,76,44,86) | ||||
(0,100,24,51,29) | (0,75,\(\infty \),31,6) | (65,41,0,10,109) | (2,0,92,35,89) | ||||
(92,0,9,37,75) | (84,75,0,56,23) | (0,52,72,19,34) | |||||
116 | (1,94,67,88,0) | (0,101,63,54,68) | (11,16,0,35,67) | (63,0,106,39,115) | mod 116 | 2668 | \(\frac{(6+2+4)\times 116}{2668}\) |
(8,0,42,65,11) | (41,56,0,87,113) | (99,5,0,16,12) | (0,35,41,20,60) | ||||
(0,62,55,99,100) | (26,0,41,114,107) | (0,37,61,25,23) | (0,4,34,102,74) | ||||
(0,64,71,18,74) | (114,0,50,82,108) | (24,0,89,99,103) | (34,0,12,98,9) | ||||
(66,68,95,74,0) | (0,47,77,90,85) | ||||||
(19,0,55,86,73) | |||||||
(12,0,106,51,71) | |||||||
(40,20,99,36,0) | |||||||
(33,69,0,1,46) | |||||||
(39,0,28,97,72) |
v | Base blocks | \(b_v\) | d | ||||
|---|---|---|---|---|---|---|---|
130 | (26,67,0,16,97) | (100,18,0,19,91) | (71,12,0,75,127) | (0,45,18,82,110) | mod 129 | 3354 | \(\frac{13\times 129}{3354}\) |
(101,36,0,95,79) | (0,82,11,21,34) | (0,12,49,110,63) | (57,122,79,0,120) | ||||
(0,54,57,59,108) | (0,83,114,11,123) | (11,0,105,111,71) | (8,26,\(\infty \),51,0) | ||||
(3,0,112,70,69) | (0,124,41,101,74) | (0,24,32,46,122) | (40,52,36,0,56) | ||||
(1,32,33,0,116) | (0,26,50,94,53) | (0,39,68,77,35) | (0,69,39,75,92) | ||||
(10,0,101,7,20) | (23,11,0,85,127) | (0,15,105,42,122) | (0,30,5,38,114) | ||||
(95,0,21,40,120) | |||||||
(8,93,0,108,52) | |||||||
131 | (0,70,1,17,59) | (0,118,2,34,9) | (68,0,4,18,105) | (8,36,5,0,79) | mod 131 | 3406 | \(\frac{13\times 131}{3406}\) |
(70,53,11,69,0) | (32,20,0,13,54) | (26,49,0,117,93) | (55,0,52,98,103) | ||||
(16,0,72,10,27) | (0,40,108,26,64) | (104,125,39,0,29) | (78,119,58,77,0) | ||||
(72,62,45,56,0) | (18,81,0,14,44) | (100,71,92,0,35) | (77,130,89,0,19) | ||||
(0,25,107,23,116) | (101,50,0,46,83) | (22,9,0,106,7) | (0,28,36,88,31) | ||||
(129,23,47,0,38) | (127,76,46,0,94) | (0,124,13,112,90) | (128,80,52,0,85) | ||||
(123,21,92,0,57) | |||||||
(0,104,65,110,75) |
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Mohammadnezhad, M., Golalizadeh, S., Boostan, M. et al. Super-Simple (v, 5, 2) Directed Designs and Their Smallest Defining Sets with Application in LDPC Codes. Bull. Iran. Math. Soc. 49, 90 (2023). https://doi.org/10.1007/s41980-023-00835-y
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DOI: https://doi.org/10.1007/s41980-023-00835-y