Best Known (136, 156, s)-Nets in Base 3
(136, 156, 17723)-Net over F3 — Constructive and digital
Digital (136, 156, 17723)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 12, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (124, 144, 17715)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 17715, F3, 20, 20) (dual of [(17715, 20), 354156, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3144, 177150, F3, 20) (dual of [177150, 177006, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 177158, F3, 20) (dual of [177158, 177014, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3144, 177158, F3, 20) (dual of [177158, 177014, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3144, 177150, F3, 20) (dual of [177150, 177006, 21]-code), using
- net defined by OOA [i] based on linear OOA(3144, 17715, F3, 20, 20) (dual of [(17715, 20), 354156, 21]-NRT-code), using
- digital (2, 12, 8)-net over F3, using
(136, 156, 75332)-Net over F3 — Digital
Digital (136, 156, 75332)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3156, 75332, F3, 2, 20) (dual of [(75332, 2), 150508, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3156, 88601, F3, 2, 20) (dual of [(88601, 2), 177046, 21]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3155, 88601, F3, 2, 20) (dual of [(88601, 2), 177047, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3155, 177202, F3, 20) (dual of [177202, 177047, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3155, 177202, F3, 20) (dual of [177202, 177047, 21]-code), using
- 31 times duplication [i] based on linear OOA(3155, 88601, F3, 2, 20) (dual of [(88601, 2), 177047, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3156, 88601, F3, 2, 20) (dual of [(88601, 2), 177046, 21]-NRT-code), using
(136, 156, large)-Net in Base 3 — Upper bound on s
There is no (136, 156, large)-net in base 3, because
- 18 times m-reduction [i] would yield (136, 138, large)-net in base 3, but