Best Known (132, 132+27, s)-Nets in Base 3
(132, 132+27, 711)-Net over F3 — Constructive and digital
Digital (132, 159, 711)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (10, 23, 23)-net over F3, using
- digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
(132, 132+27, 5249)-Net over F3 — Digital
Digital (132, 159, 5249)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 5249, F3, 27) (dual of [5249, 5090, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 6614, F3, 27) (dual of [6614, 6455, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3159, 6614, F3, 27) (dual of [6614, 6455, 28]-code), using
(132, 132+27, 1783328)-Net in Base 3 — Upper bound on s
There is no (132, 159, 1783329)-net in base 3, because
- 1 times m-reduction [i] would yield (132, 158, 1783329)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2427 494959 006199 906474 169723 841899 968701 253136 116508 494565 272423 922652 883475 > 3158 [i]