Best Known (209−39, 209, s)-Nets in Base 3
(209−39, 209, 896)-Net over F3 — Constructive and digital
Digital (170, 209, 896)-net over F3, using
- 31 times duplication [i] based on digital (169, 208, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 52, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 52, 224)-net over F81, using
(209−39, 209, 3489)-Net over F3 — Digital
Digital (170, 209, 3489)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3209, 3489, F3, 39) (dual of [3489, 3280, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using
(209−39, 209, 662819)-Net in Base 3 — Upper bound on s
There is no (170, 209, 662820)-net in base 3, because
- 1 times m-reduction [i] would yield (170, 208, 662820)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1742 697806 691931 017538 292134 989719 529133 071144 498449 531518 456895 149608 226073 974450 988782 685674 807345 > 3208 [i]