Best Known (112, 157, s)-Nets in Base 8
(112, 157, 1026)-Net over F8 — Constructive and digital
Digital (112, 157, 1026)-net over F8, using
- 11 times m-reduction [i] based on digital (112, 168, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 84, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 84, 513)-net over F64, using
(112, 157, 4136)-Net over F8 — Digital
Digital (112, 157, 4136)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8157, 4136, F8, 45) (dual of [4136, 3979, 46]-code), using
- 3978 step Varšamov–Edel lengthening with (ri) = (7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 36 times 0, 1, 38 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 54 times 0, 1, 56 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 69 times 0, 1, 72 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 92 times 0, 1, 96 times 0, 1, 100 times 0, 1, 106 times 0, 1, 111 times 0, 1, 116 times 0, 1, 122 times 0, 1, 128 times 0, 1, 135 times 0, 1, 141 times 0, 1, 148 times 0, 1, 155 times 0, 1, 163 times 0, 1, 171 times 0, 1, 179 times 0, 1, 188 times 0) [i] based on linear OA(845, 46, F8, 45) (dual of [46, 1, 46]-code or 46-arc in PG(44,8)), using
- dual of repetition code with length 46 [i]
- 3978 step Varšamov–Edel lengthening with (ri) = (7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 36 times 0, 1, 38 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 54 times 0, 1, 56 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 69 times 0, 1, 72 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 92 times 0, 1, 96 times 0, 1, 100 times 0, 1, 106 times 0, 1, 111 times 0, 1, 116 times 0, 1, 122 times 0, 1, 128 times 0, 1, 135 times 0, 1, 141 times 0, 1, 148 times 0, 1, 155 times 0, 1, 163 times 0, 1, 171 times 0, 1, 179 times 0, 1, 188 times 0) [i] based on linear OA(845, 46, F8, 45) (dual of [46, 1, 46]-code or 46-arc in PG(44,8)), using
(112, 157, 3277043)-Net in Base 8 — Upper bound on s
There is no (112, 157, 3277044)-net in base 8, because
- 1 times m-reduction [i] would yield (112, 156, 3277044)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 762 149310 881100 075724 649526 294352 195027 635829 225709 948406 287143 076162 262083 330925 942875 969848 168743 326799 828978 618092 306082 923085 016732 818616 > 8156 [i]