Best Known (165−21, 165, s)-Nets in Base 3
(165−21, 165, 17721)-Net over F3 — Constructive and digital
Digital (144, 165, 17721)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (133, 154, 17714)-net over F3, using
- net defined by OOA [i] based on linear OOA(3154, 17714, F3, 21, 21) (dual of [(17714, 21), 371840, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3154, 177141, F3, 21) (dual of [177141, 176987, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 177146, F3, 21) (dual of [177146, 176992, 22]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 177146, F3, 21) (dual of [177146, 176992, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3154, 177141, F3, 21) (dual of [177141, 176987, 22]-code), using
- net defined by OOA [i] based on linear OOA(3154, 17714, F3, 21, 21) (dual of [(17714, 21), 371840, 22]-NRT-code), using
- digital (1, 11, 7)-net over F3, using
(165−21, 165, 78996)-Net over F3 — Digital
Digital (144, 165, 78996)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3165, 78996, F3, 2, 21) (dual of [(78996, 2), 157827, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3165, 88600, F3, 2, 21) (dual of [(88600, 2), 177035, 22]-NRT-code), using
- 1 step truncation [i] based on linear OOA(3166, 88601, F3, 2, 22) (dual of [(88601, 2), 177036, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3166, 177202, F3, 22) (dual of [177202, 177036, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [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(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(21) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3166, 177202, F3, 22) (dual of [177202, 177036, 23]-code), using
- 1 step truncation [i] based on linear OOA(3166, 88601, F3, 2, 22) (dual of [(88601, 2), 177036, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3165, 88600, F3, 2, 21) (dual of [(88600, 2), 177035, 22]-NRT-code), using
(165−21, 165, large)-Net in Base 3 — Upper bound on s
There is no (144, 165, large)-net in base 3, because
- 19 times m-reduction [i] would yield (144, 146, large)-net in base 3, but