Best Known (68, 68+31, s)-Nets in Base 8
(68, 68+31, 389)-Net over F8 — Constructive and digital
Digital (68, 99, 389)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (8, 23, 35)-net over F8, using
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 7, N(F) = 34, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 7 and N(F) ≥ 34, using a function field by Sémirat [i]
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- digital (45, 76, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 177)-net over F64, using
- digital (8, 23, 35)-net over F8, using
(68, 68+31, 576)-Net in Base 8 — Constructive
(68, 99, 576)-net in base 8, using
- 3 times m-reduction [i] based on (68, 102, 576)-net in base 8, using
- trace code for nets [i] based on (17, 51, 288)-net in base 64, using
- 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 48, 288)-net over F128, using
- 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- trace code for nets [i] based on (17, 51, 288)-net in base 64, using
(68, 68+31, 1659)-Net over F8 — Digital
Digital (68, 99, 1659)-net over F8, using
(68, 68+31, 729207)-Net in Base 8 — Upper bound on s
There is no (68, 99, 729208)-net in base 8, because
- 1 times m-reduction [i] would yield (68, 98, 729208)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 31829 076276 925268 576151 603837 993732 042220 968640 593762 967515 482906 217099 766645 191572 079226 > 898 [i]