Best Known (105−40, 105, s)-Nets in Base 8
(105−40, 105, 354)-Net over F8 — Constructive and digital
Digital (65, 105, 354)-net over F8, using
- 11 times m-reduction [i] based on digital (65, 116, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 177)-net over F64, using
(105−40, 105, 384)-Net in Base 8 — Constructive
(65, 105, 384)-net in base 8, using
- 3 times m-reduction [i] based on (65, 108, 384)-net in base 8, using
- trace code for nets [i] based on (11, 54, 192)-net in base 64, using
- 2 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
- base change [i] based on digital (3, 48, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 48, 192)-net over F128, using
- 2 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
- trace code for nets [i] based on (11, 54, 192)-net in base 64, using
(105−40, 105, 614)-Net over F8 — Digital
Digital (65, 105, 614)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8105, 614, F8, 40) (dual of [614, 509, 41]-code), using
- 508 step Varšamov–Edel lengthening with (ri) = (7, 2, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 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, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 25 times 0, 1, 25 times 0, 1, 28 times 0, 1, 29 times 0) [i] based on linear OA(840, 41, F8, 40) (dual of [41, 1, 41]-code or 41-arc in PG(39,8)), using
- dual of repetition code with length 41 [i]
- 508 step Varšamov–Edel lengthening with (ri) = (7, 2, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 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, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 25 times 0, 1, 25 times 0, 1, 28 times 0, 1, 29 times 0) [i] based on linear OA(840, 41, F8, 40) (dual of [41, 1, 41]-code or 41-arc in PG(39,8)), using
(105−40, 105, 65365)-Net in Base 8 — Upper bound on s
There is no (65, 105, 65366)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 66759 972825 977381 793681 499678 788690 309884 891571 921072 202316 487081 863165 777541 197238 992684 993873 > 8105 [i]