Best Known (2, 2+9, s)-Nets in Base 8
(2, 2+9, 17)-Net over F8 — Constructive and digital
Digital (2, 11, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
(2, 2+9, 18)-Net over F8 — Digital
Digital (2, 11, 18)-net over F8, using
- net from sequence [i] based on digital (2, 17)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 18, using
(2, 2+9, 54)-Net over F8 — Upper bound on s (digital)
There is no digital (2, 11, 55)-net over F8, because
- 1 times m-reduction [i] would yield digital (2, 10, 55)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(810, 55, F8, 8) (dual of [55, 45, 9]-code), but
- construction Y1 [i] would yield
- linear OA(89, 15, F8, 8) (dual of [15, 6, 9]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- OA(845, 55, S8, 40), but
- discarding factors would yield OA(845, 52, S8, 40), but
- the linear programming bound shows that M ≥ 17082 370822 074457 326416 360196 848644 913580 998656 / 353625 > 845 [i]
- discarding factors would yield OA(845, 52, S8, 40), but
- linear OA(89, 15, F8, 8) (dual of [15, 6, 9]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(810, 55, F8, 8) (dual of [55, 45, 9]-code), but
(2, 2+9, 55)-Net in Base 8 — Upper bound on s
There is no (2, 11, 56)-net in base 8, because
- 1 times m-reduction [i] would yield (2, 10, 56)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1135 562751 > 810 [i]