Best Known (121−45, 121, s)-Nets in Base 8
(121−45, 121, 354)-Net over F8 — Constructive and digital
Digital (76, 121, 354)-net over F8, using
- 17 times m-reduction [i] based on digital (76, 138, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 69, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 69, 177)-net over F64, using
(121−45, 121, 514)-Net in Base 8 — Constructive
(76, 121, 514)-net in base 8, using
- 81 times duplication [i] based on (75, 120, 514)-net in base 8, using
- base change [i] based on digital (45, 90, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 45, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 45, 257)-net over F256, using
- base change [i] based on digital (45, 90, 514)-net over F16, using
(121−45, 121, 773)-Net over F8 — Digital
Digital (76, 121, 773)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8121, 773, F8, 45) (dual of [773, 652, 46]-code), using
- 651 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) [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]
- 651 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) [i] based on linear OA(845, 46, F8, 45) (dual of [46, 1, 46]-code or 46-arc in PG(44,8)), using
(121−45, 121, 109055)-Net in Base 8 — Upper bound on s
There is no (76, 121, 109056)-net in base 8, because
- 1 times m-reduction [i] would yield (76, 120, 109056)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2 348961 208831 102836 267430 836805 515189 053426 307648 412761 315243 591858 592271 696682 115247 441631 842973 046600 526753 > 8120 [i]