Best Known (149−29, 149, s)-Nets in Base 3
(149−29, 149, 688)-Net over F3 — Constructive and digital
Digital (120, 149, 688)-net over F3, using
- 31 times duplication [i] based on digital (119, 148, 688)-net over F3, using
- t-expansion [i] based on digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 172)-net over F81, using
- t-expansion [i] based on digital (118, 148, 688)-net over F3, using
(149−29, 149, 2227)-Net over F3 — Digital
Digital (120, 149, 2227)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3149, 2227, F3, 29) (dual of [2227, 2078, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3149, 2237, F3, 29) (dual of [2237, 2088, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3148, 2236, F3, 29) (dual of [2236, 2088, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(314, 49, F3, 6) (dual of [49, 35, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3148, 2236, F3, 29) (dual of [2236, 2088, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3149, 2237, F3, 29) (dual of [2237, 2088, 30]-code), using
(149−29, 149, 334397)-Net in Base 3 — Upper bound on s
There is no (120, 149, 334398)-net in base 3, because
- 1 times m-reduction [i] would yield (120, 148, 334398)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 41110 090926 776859 341980 035766 034134 798750 824872 772546 189603 854514 703029 > 3148 [i]