Best Known (33−24, 33, s)-Nets in Base 4
(33−24, 33, 22)-Net over F4 — Constructive and digital
Digital (9, 33, 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
(33−24, 33, 26)-Net over F4 — Digital
Digital (9, 33, 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
(33−24, 33, 63)-Net over F4 — Upper bound on s (digital)
There is no digital (9, 33, 64)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(433, 64, F4, 24) (dual of [64, 31, 25]-code), but
- residual code [i] would yield OA(49, 39, S4, 6), but
- the linear programming bound shows that M ≥ 504 832000 / 1843 > 49 [i]
- residual code [i] would yield OA(49, 39, S4, 6), but
(33−24, 33, 70)-Net in Base 4 — Upper bound on s
There is no (9, 33, 71)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 77 190653 118924 257176 > 433 [i]
- extracting embedded orthogonal array [i] would yield OA(433, 71, S4, 24), but
- the linear programming bound shows that M ≥ 59723 916380 964784 140612 335617 515793 274816 846398 594188 574720 / 731 833994 497895 449624 685241 072880 229647 > 433 [i]