Best Known (126, 126+13, s)-Nets in Base 3
(126, 126+13, 1398184)-Net over F3 — Constructive and digital
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
(126, 126+13, 4194411)-Net over F3 — Digital
Digital (126, 139, 4194411)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3139, 4194411, F3, 2, 13) (dual of [(4194411, 2), 8388683, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(318, 110, F3, 2, 6) (dual of [(110, 2), 202, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(318, 110, F3, 6) (dual of [110, 92, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(317, 108, F3, 6) (dual of [108, 91, 7]-code), using
- linear OA(317, 109, F3, 5) (dual of [109, 92, 6]-code), using Gilbert–Varšamov bound and bm = 317 > Vbs−1(k−1) = 87 402097 [i]
- linear OA(30, 1, F3, 0) (dual of [1, 1, 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 with Varšamov bound [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(318, 110, F3, 6) (dual of [110, 92, 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(318, 110, F3, 2, 6) (dual of [(110, 2), 202, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(126, 126+13, large)-Net in Base 3 — Upper bound on s
There is no (126, 139, large)-net in base 3, because
- 11 times m-reduction [i] would yield (126, 128, large)-net in base 3, but