Best Known (149, 149+40, s)-Nets in Base 3
(149, 149+40, 688)-Net over F3 — Constructive and digital
Digital (149, 189, 688)-net over F3, using
- 31 times duplication [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
(149, 149+40, 1688)-Net over F3 — Digital
Digital (149, 189, 1688)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3189, 1688, F3, 40) (dual of [1688, 1499, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3189, 2208, F3, 40) (dual of [2208, 2019, 41]-code), using
- construction XX applied to Ce(39) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(34, 19, F3, 2) (dual of [19, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(39) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3189, 2208, F3, 40) (dual of [2208, 2019, 41]-code), using
(149, 149+40, 133970)-Net in Base 3 — Upper bound on s
There is no (149, 189, 133971)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 499530 743774 528563 154865 740746 210899 016751 797317 673982 701940 824779 247943 926409 910573 558201 > 3189 [i]