Best Known (40−31, 40, s)-Nets in Base 4
(40−31, 40, 22)-Net over F4 — Constructive and digital
Digital (9, 40, 22)-net over F4, using
- net from sequence [i] based on digital (9, 21)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 9 and N(F) ≥ 22, using
(40−31, 40, 26)-Net over F4 — Digital
Digital (9, 40, 26)-net over F4, using
- net from sequence [i] based on digital (9, 25)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 9 and N(F) ≥ 26, using
(40−31, 40, 44)-Net over F4 — Upper bound on s (digital)
There is no digital (9, 40, 45)-net over F4, because
- 1 times m-reduction [i] would yield digital (9, 39, 45)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(439, 45, F4, 30) (dual of [45, 6, 31]-code), but
(40−31, 40, 45)-Net in Base 4 — Upper bound on s
There is no (9, 40, 46)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(440, 46, S4, 31), but
- the linear programming bound shows that M ≥ 976 812062 248620 373162 590208 / 703 > 440 [i]