Best Known (68−63, 68, s)-Nets in Base 8
(68−63, 68, 28)-Net over F8 — Constructive and digital
Digital (5, 68, 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
(68−63, 68, 29)-Net over F8 — Digital
Digital (5, 68, 29)-net over F8, using
- net from sequence [i] based on digital (5, 28)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 29, using
(68−63, 68, 49)-Net over F8 — Upper bound on s (digital)
There is no digital (5, 68, 50)-net over F8, because
- 23 times m-reduction [i] would yield digital (5, 45, 50)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(845, 50, F8, 40) (dual of [50, 5, 41]-code), but
- construction Y1 [i] would yield
- OA(844, 46, S8, 40), but
- the (dual) Plotkin bound shows that M ≥ 261336 857795 280739 939871 698507 597986 398208 / 41 > 844 [i]
- OA(85, 50, S8, 4), but
- discarding factors would yield OA(85, 37, S8, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 32894 > 85 [i]
- discarding factors would yield OA(85, 37, S8, 4), but
- OA(844, 46, S8, 40), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(845, 50, F8, 40) (dual of [50, 5, 41]-code), but
(68−63, 68, 51)-Net in Base 8 — Upper bound on s
There is no (5, 68, 52)-net in base 8, because
- 23 times m-reduction [i] would yield (5, 45, 52)-net in base 8, but
- extracting embedded orthogonal array [i] 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]
- extracting embedded orthogonal array [i] would yield OA(845, 52, S8, 40), but