Best Known (157, 157+17, s)-Nets in Base 3
(157, 157+17, 1048579)-Net over F3 — Constructive and digital
Digital (157, 174, 1048579)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (149, 166, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- digital (0, 8, 4)-net over F3, using
(157, 157+17, 2796205)-Net over F3 — Digital
Digital (157, 174, 2796205)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3174, 2796205, F3, 3, 17) (dual of [(2796205, 3), 8388441, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(38, 4, F3, 3, 8) (dual of [(4, 3), 4, 9]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;4,3) [i]
- linear OOA(3166, 2796201, F3, 3, 17) (dual of [(2796201, 3), 8388437, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- OOA 3-folding [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- linear OOA(38, 4, F3, 3, 8) (dual of [(4, 3), 4, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(157, 157+17, large)-Net in Base 3 — Upper bound on s
There is no (157, 174, large)-net in base 3, because
- 15 times m-reduction [i] would yield (157, 159, large)-net in base 3, but