Best Known (168−23, 168, s)-Nets in Base 3
(168−23, 168, 16105)-Net over F3 — Constructive and digital
Digital (145, 168, 16105)-net over F3, using
- 32 times duplication [i] based on digital (143, 166, 16105)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 16105, F3, 23, 23) (dual of [(16105, 23), 370249, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3166, 177156, F3, 23) (dual of [177156, 176990, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 177158, F3, 23) (dual of [177158, 176992, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 177158, F3, 23) (dual of [177158, 176992, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3166, 177156, F3, 23) (dual of [177156, 176990, 24]-code), using
- net defined by OOA [i] based on linear OOA(3166, 16105, F3, 23, 23) (dual of [(16105, 23), 370249, 24]-NRT-code), using
(168−23, 168, 55140)-Net over F3 — Digital
Digital (145, 168, 55140)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3168, 55140, F3, 3, 23) (dual of [(55140, 3), 165252, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3168, 59053, F3, 3, 23) (dual of [(59053, 3), 176991, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3167, 59053, F3, 3, 23) (dual of [(59053, 3), 176992, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3167, 177159, F3, 23) (dual of [177159, 176992, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3166, 177158, F3, 23) (dual of [177158, 176992, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3166, 177158, F3, 23) (dual of [177158, 176992, 24]-code), using
- OOA 3-folding [i] based on linear OA(3167, 177159, F3, 23) (dual of [177159, 176992, 24]-code), using
- 31 times duplication [i] based on linear OOA(3167, 59053, F3, 3, 23) (dual of [(59053, 3), 176992, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3168, 59053, F3, 3, 23) (dual of [(59053, 3), 176991, 24]-NRT-code), using
(168−23, 168, large)-Net in Base 3 — Upper bound on s
There is no (145, 168, large)-net in base 3, because
- 21 times m-reduction [i] would yield (145, 147, large)-net in base 3, but