Best Known (68, 68+37, s)-Nets in Base 9
(68, 68+37, 448)-Net over F9 — Constructive and digital
Digital (68, 105, 448)-net over F9, using
- 5 times m-reduction [i] based on digital (68, 110, 448)-net over F9, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
(68, 68+37, 1102)-Net over F9 — Digital
Digital (68, 105, 1102)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9105, 1102, F9, 37) (dual of [1102, 997, 38]-code), using
- 996 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 1, 1, 1, 1, 1, 0, 1, 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, 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, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 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, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 55 times 0, 1, 58 times 0, 1, 63 times 0) [i] based on linear OA(937, 38, F9, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,9)), using
- dual of repetition code with length 38 [i]
- 996 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 1, 1, 1, 1, 1, 0, 1, 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, 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, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 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, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 55 times 0, 1, 58 times 0, 1, 63 times 0) [i] based on linear OA(937, 38, F9, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,9)), using
(68, 68+37, 307911)-Net in Base 9 — Upper bound on s
There is no (68, 105, 307912)-net in base 9, because
- 1 times m-reduction [i] would yield (68, 104, 307912)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 1742 732472 083227 891575 577877 872176 144073 802662 859814 437679 915753 061710 494617 915166 263943 842560 051585 > 9104 [i]