Best Known (218−19, 218, s)-Nets in Base 2
(218−19, 218, 932071)-Net over F2 — Constructive and digital
Digital (199, 218, 932071)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 5)-net over F2, using
- net from sequence [i] based on digital (1, 4)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 1 and N(F) ≥ 5, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (1, 4)-sequence over F2, using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- digital (1, 10, 5)-net over F2, using
(218−19, 218, 1198376)-Net over F2 — Digital
Digital (199, 218, 1198376)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2218, 1198376, F2, 7, 19) (dual of [(1198376, 7), 8388414, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 5, F2, 7, 9) (dual of [(5, 7), 25, 10]-NRT-code), using
- extended algebraic-geometric NRT-code AGe(7;F,25P) [i] based on function field F/F2 with g(F) = 1 and N(F) ≥ 5, using
- linear OOA(2208, 1198371, F2, 7, 19) (dual of [(1198371, 7), 8388389, 20]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2208, 8388597, F2, 19) (dual of [8388597, 8388389, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 7-folding [i] based on linear OA(2208, 8388597, F2, 19) (dual of [8388597, 8388389, 20]-code), using
- linear OOA(210, 5, F2, 7, 9) (dual of [(5, 7), 25, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(218−19, 218, large)-Net in Base 2 — Upper bound on s
There is no (199, 218, large)-net in base 2, because
- 17 times m-reduction [i] would yield (199, 201, large)-net in base 2, but