Best Known (81−75, 81, s)-Nets in Base 8
(81−75, 81, 28)-Net over F8 — Constructive and digital
Digital (6, 81, 28)-net over F8, using
- t-expansion [i] based on digital (5, 81, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
(81−75, 81, 33)-Net over F8 — Digital
Digital (6, 81, 33)-net over F8, using
- net from sequence [i] based on digital (6, 32)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 6 and N(F) ≥ 33, using
(81−75, 81, 57)-Net over F8 — Upper bound on s (digital)
There is no digital (6, 81, 58)-net over F8, because
- 33 times m-reduction [i] would yield digital (6, 48, 58)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(848, 58, F8, 42) (dual of [58, 10, 43]-code), but
- construction Y1 [i] would yield
- linear OA(847, 50, F8, 42) (dual of [50, 3, 43]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- OA(810, 58, S8, 8), but
- discarding factors would yield OA(810, 57, S8, 8), but
- the linear programming bound shows that M ≥ 1426 626655 027200 / 1 315507 > 810 [i]
- discarding factors would yield OA(810, 57, S8, 8), but
- linear OA(847, 50, F8, 42) (dual of [50, 3, 43]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(848, 58, F8, 42) (dual of [58, 10, 43]-code), but
(81−75, 81, 59)-Net in Base 8 — Upper bound on s
There is no (6, 81, 60)-net in base 8, because
- 28 times m-reduction [i] would yield (6, 53, 60)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(853, 60, S8, 47), but
- the linear programming bound shows that M ≥ 10809 266109 699357 983034 453022 769629 213375 277087 850496 / 13091 > 853 [i]
- extracting embedded orthogonal array [i] would yield OA(853, 60, S8, 47), but