Best Known (127, 127+13, s)-Nets in Base 3
(127, 127+13, 1398184)-Net over F3 — Constructive and digital
Digital (127, 140, 1398184)-net over F3, using
- 31 times duplication [i] based on digital (126, 139, 1398184)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (12, 18, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- trace code for nets [i] based on digital (0, 6, 28)-net over F27, using
- digital (108, 121, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- digital (12, 18, 84)-net over F3, using
- (u, u+v)-construction [i] based on
(127, 127+13, 4194422)-Net over F3 — Digital
Digital (127, 140, 4194422)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3140, 4194422, F3, 2, 13) (dual of [(4194422, 2), 8388704, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(319, 121, F3, 2, 6) (dual of [(121, 2), 223, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(319, 121, F3, 6) (dual of [121, 102, 7]-code), using
- 1 times truncation [i] based on linear OA(320, 122, F3, 7) (dual of [122, 102, 8]-code), using
- a “DaH†code from Brouwer’s database [i]
- 1 times truncation [i] based on linear OA(320, 122, F3, 7) (dual of [122, 102, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(319, 121, F3, 6) (dual of [121, 102, 7]-code), using
- linear OOA(3121, 4194301, F3, 2, 13) (dual of [(4194301, 2), 8388481, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3121, 8388602, F3, 13) (dual of [8388602, 8388481, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 2-folding [i] based on linear OA(3121, 8388602, F3, 13) (dual of [8388602, 8388481, 14]-code), using
- linear OOA(319, 121, F3, 2, 6) (dual of [(121, 2), 223, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(127, 127+13, large)-Net in Base 3 — Upper bound on s
There is no (127, 140, large)-net in base 3, because
- 11 times m-reduction [i] would yield (127, 129, large)-net in base 3, but