Best Known (136, 136+22, s)-Nets in Base 3
(136, 136+22, 16105)-Net over F3 — Constructive and digital
Digital (136, 158, 16105)-net over F3, using
- 32 times duplication [i] based on digital (134, 156, 16105)-net over F3, using
- net defined by OOA [i] based on linear OOA(3156, 16105, F3, 22, 22) (dual of [(16105, 22), 354154, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3156, 177155, F3, 22) (dual of [177155, 176999, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3156, 177155, F3, 22) (dual of [177155, 176999, 23]-code), using
- net defined by OOA [i] based on linear OOA(3156, 16105, F3, 22, 22) (dual of [(16105, 22), 354154, 23]-NRT-code), using
(136, 136+22, 48472)-Net over F3 — Digital
Digital (136, 158, 48472)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3158, 48472, F3, 3, 22) (dual of [(48472, 3), 145258, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3158, 59053, F3, 3, 22) (dual of [(59053, 3), 177001, 23]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3156, 59053, F3, 3, 22) (dual of [(59053, 3), 177003, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- 32 times duplication [i] based on linear OOA(3156, 59053, F3, 3, 22) (dual of [(59053, 3), 177003, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3158, 59053, F3, 3, 22) (dual of [(59053, 3), 177001, 23]-NRT-code), using
(136, 136+22, large)-Net in Base 3 — Upper bound on s
There is no (136, 158, large)-net in base 3, because
- 20 times m-reduction [i] would yield (136, 138, large)-net in base 3, but