Best Known (127, 127+12, s)-Nets in Base 3
(127, 127+12, 1398184)-Net over F3 — Constructive and digital
Digital (127, 139, 1398184)-net over F3, using
- t-expansion [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+12, large)-Net over F3 — Digital
Digital (127, 139, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
(127, 127+12, large)-Net in Base 3 — Upper bound on s
There is no (127, 139, large)-net in base 3, because
- 10 times m-reduction [i] would yield (127, 129, large)-net in base 3, but