Best Known (40−8, 40, s)-Nets in Base 3
(40−8, 40, 552)-Net over F3 — Constructive and digital
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
(40−8, 40, 1885)-Net over F3 — Digital
Digital (32, 40, 1885)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 1885, F3, 8) (dual of [1885, 1845, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 2192, F3, 8) (dual of [2192, 2152, 9]-code), using
- (u, u+v)-construction [i] based on
- linear OA(34, 5, F3, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,3)), using
- dual of repetition code with length 5 [i]
- 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(34, 5, F3, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(340, 2192, F3, 8) (dual of [2192, 2152, 9]-code), using
(40−8, 40, 65344)-Net in Base 3 — Upper bound on s
There is no (32, 40, 65345)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 12 157694 664356 231761 > 340 [i]