Best Known (219, 219+14, s)-Nets in Base 2
(219, 219+14, 2396748)-Net over F2 — Constructive and digital
Digital (219, 233, 2396748)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 65, 2097150)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(265, 2097151, F2, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- digital (154, 168, 1198374)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 3)-net over F2, using
- net from sequence [i] based on digital (0, 2)-sequence over F2, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 0 and N(F) ≥ 3, using
- the rational function field F2(x) [i]
- Niederreiter sequence [i]
- Sobol sequence [i]
- net from sequence [i] based on digital (0, 2)-sequence over F2, using
- digital (147, 161, 1198371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2161, 1198371, F2, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2161, 8388597, F2, 14) (dual of [8388597, 8388436, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2161, 8388597, F2, 14) (dual of [8388597, 8388436, 15]-code), using
- net defined by OOA [i] based on linear OOA(2161, 1198371, F2, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
- digital (0, 7, 3)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 65, 2097150)-net over F2, using
(219, 219+14, large)-Net over F2 — Digital
Digital (219, 233, large)-net over F2, using
- 22 times duplication [i] based on digital (217, 231, large)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, large, F2, 2, 14), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2229, 8388602, F2, 2, 14) (dual of [(8388602, 2), 16776975, 15]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(268, 4194303, F2, 2, 7) (dual of [(4194303, 2), 8388538, 8]-NRT-code), using
- linear OOA(2161, 4194301, F2, 2, 14) (dual of [(4194301, 2), 8388441, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2161, 8388602, F2, 14) (dual of [8388602, 8388441, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OOA 2-folding [i] based on linear OA(2161, 8388602, F2, 14) (dual of [8388602, 8388441, 15]-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2229, 8388602, F2, 2, 14) (dual of [(8388602, 2), 16776975, 15]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, large, F2, 2, 14), using
(219, 219+14, large)-Net in Base 2 — Upper bound on s
There is no (219, 233, large)-net in base 2, because
- 12 times m-reduction [i] would yield (219, 221, large)-net in base 2, but