Best Known (38−33, 38, s)-Nets in Base 8
(38−33, 38, 28)-Net over F8 — Constructive and digital
Digital (5, 38, 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
(38−33, 38, 29)-Net over F8 — Digital
Digital (5, 38, 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
(38−33, 38, 67)-Net over F8 — Upper bound on s (digital)
There is no digital (5, 38, 68)-net over F8, because
- 1 times m-reduction [i] would yield digital (5, 37, 68)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(837, 68, F8, 32) (dual of [68, 31, 33]-code), but
- residual code [i] would yield OA(85, 35, S8, 4), but
- the linear programming bound shows that M ≥ 1 306624 / 39 > 85 [i]
- residual code [i] would yield OA(85, 35, S8, 4), but
- extracting embedded orthogonal array [i] would yield linear OA(837, 68, F8, 32) (dual of [68, 31, 33]-code), but
(38−33, 38, 86)-Net in Base 8 — Upper bound on s
There is no (5, 38, 87)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(838, 87, S8, 33), but
- the linear programming bound shows that M ≥ 1 205787 875960 632895 111659 667315 586273 528594 165947 266312 394680 616880 177152 / 57 443164 359168 229027 114416 485774 529519 > 838 [i]