Best Known (186−37, 186, s)-Nets in Base 3
(186−37, 186, 688)-Net over F3 — Constructive and digital
Digital (149, 186, 688)-net over F3, using
- t-expansion [i] based on digital (148, 186, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (148, 188, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 47, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 47, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (148, 188, 688)-net over F3, using
(186−37, 186, 2280)-Net over F3 — Digital
Digital (149, 186, 2280)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3186, 2280, F3, 37) (dual of [2280, 2094, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 2291, F3, 37) (dual of [2291, 2105, 38]-code), using
- 86 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0, 1, 20 times 0) [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 86 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0, 1, 20 times 0) [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 2291, F3, 37) (dual of [2291, 2105, 38]-code), using
(186−37, 186, 302566)-Net in Base 3 — Upper bound on s
There is no (149, 186, 302567)-net in base 3, because
- 1 times m-reduction [i] would yield (149, 185, 302567)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 18511 392244 266227 726379 418948 152156 221874 926456 856353 785413 571153 376158 852697 083867 499565 > 3185 [i]