Best Known (183−39, 183, s)-Nets in Base 3
(183−39, 183, 640)-Net over F3 — Constructive and digital
Digital (144, 183, 640)-net over F3, using
- t-expansion [i] based on digital (143, 183, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (143, 184, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 46, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 46, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (143, 184, 640)-net over F3, using
(183−39, 183, 1595)-Net over F3 — Digital
Digital (144, 183, 1595)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3183, 1595, F3, 39) (dual of [1595, 1412, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3183, 2188, F3, 39) (dual of [2188, 2005, 40]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, 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(3183, 2188, F3, 39) (dual of [2188, 2005, 40]-code), using
(183−39, 183, 147384)-Net in Base 3 — Upper bound on s
There is no (144, 183, 147385)-net in base 3, because
- 1 times m-reduction [i] would yield (144, 182, 147385)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 685 682964 934696 110400 287127 057500 768976 651798 078615 287406 798207 815608 384879 050467 737211 > 3182 [i]