Best Known (189, 189+17, s)-Nets in Base 3
(189, 189+17, 1049127)-Net over F3 — Constructive and digital
Digital (189, 206, 1049127)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (32, 40, 552)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (28, 36, 548)-net over F3, using
- net defined by OOA [i] based on linear OOA(336, 548, F3, 8, 8) (dual of [(548, 8), 4348, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(336, 2192, F3, 8) (dual of [2192, 2156, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(336, 2194, F3, 8) (dual of [2194, 2158, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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 applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(336, 2194, F3, 8) (dual of [2194, 2158, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(336, 2192, F3, 8) (dual of [2192, 2156, 9]-code), using
- net defined by OOA [i] based on linear OOA(336, 548, F3, 8, 8) (dual of [(548, 8), 4348, 9]-NRT-code), using
- digital (0, 4, 4)-net over F3, using
- (u, u+v)-construction [i] based on
- 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 (32, 40, 552)-net over F3, using
(189, 189+17, large)-Net over F3 — Digital
Digital (189, 206, large)-net over F3, using
- 33 times duplication [i] based on digital (186, 203, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 37 times code embedding in larger space [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]
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
(189, 189+17, large)-Net in Base 3 — Upper bound on s
There is no (189, 206, large)-net in base 3, because
- 15 times m-reduction [i] would yield (189, 191, large)-net in base 3, but