Best Known (34, 34+6, s)-Nets in Base 3
(34, 34+6, 19683)-Net over F3 — Constructive and digital
Digital (34, 40, 19683)-net over F3, using
- net defined by OOA [i] based on linear OOA(340, 19683, F3, 6, 6) (dual of [(19683, 6), 118058, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(340, 19683, F3, 5, 6) (dual of [(19683, 5), 98375, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(340, 59049, F3, 6) (dual of [59049, 59009, 7]-code), using
- 1 times truncation [i] based on linear OA(341, 59050, F3, 7) (dual of [59050, 59009, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(341, 59050, F3, 7) (dual of [59050, 59009, 8]-code), using
- OA 3-folding and stacking [i] based on linear OA(340, 59049, F3, 6) (dual of [59049, 59009, 7]-code), using
- appending kth column [i] based on linear OOA(340, 19683, F3, 5, 6) (dual of [(19683, 5), 98375, 7]-NRT-code), using
(34, 34+6, 49651)-Net over F3 — Digital
Digital (34, 40, 49651)-net over F3, using
- net defined by OOA [i] based on linear OOA(340, 49651, F3, 6, 6) (dual of [(49651, 6), 297866, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(340, 49651, F3, 5, 6) (dual of [(49651, 5), 248215, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 49651, F3, 6) (dual of [49651, 49611, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 59049, F3, 6) (dual of [59049, 59009, 7]-code), using
- 1 times truncation [i] based on linear OA(341, 59050, F3, 7) (dual of [59050, 59009, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(341, 59050, F3, 7) (dual of [59050, 59009, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 59049, F3, 6) (dual of [59049, 59009, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 49651, F3, 6) (dual of [49651, 49611, 7]-code), using
- appending kth column [i] based on linear OOA(340, 49651, F3, 5, 6) (dual of [(49651, 5), 248215, 7]-NRT-code), using
(34, 34+6, 2089151)-Net in Base 3 — Upper bound on s
There is no (34, 40, 2089152)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 12 157671 727954 471681 > 340 [i]