Best Known (172−15, 172, s)-Nets in Base 3
(172−15, 172, 1198455)-Net over F3 — Constructive and digital
Digital (157, 172, 1198455)-net over F3, using
- 31 times duplication [i] based on digital (156, 171, 1198455)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (14, 21, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 28)-net over F27, using
- digital (135, 150, 1198371)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- digital (14, 21, 84)-net over F3, using
- (u, u+v)-construction [i] based on
(172−15, 172, 5350012)-Net over F3 — Digital
Digital (157, 172, 5350012)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3172, 5350012, F3, 15) (dual of [5350012, 5349840, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3172, large, F3, 15) (dual of [large, large−172, 16]-code), using
- 22 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 22 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3172, large, F3, 15) (dual of [large, large−172, 16]-code), using
(172−15, 172, large)-Net in Base 3 — Upper bound on s
There is no (157, 172, large)-net in base 3, because
- 13 times m-reduction [i] would yield (157, 159, large)-net in base 3, but