Best Known (249−12, 249, s)-Nets in Base 2
(249−12, 249, 4194304)-Net over F2 — Constructive and digital
Digital (237, 249, 4194304)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (231, 243, 4194301)-net over F2, using
- net defined by OOA [i] based on linear OOA(2243, 4194301, F2, 15, 12) (dual of [(4194301, 15), 62914272, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2243, large, F2, 3, 12), using
- trace code [i] based on linear OOA(881, 2796201, F8, 3, 12) (dual of [(2796201, 3), 8388522, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(881, large, F8, 12) (dual of [large, large−81, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(881, large, F8, 12) (dual of [large, large−81, 13]-code), using
- trace code [i] based on linear OOA(881, 2796201, F8, 3, 12) (dual of [(2796201, 3), 8388522, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2243, large, F2, 3, 12), using
- net defined by OOA [i] based on linear OOA(2243, 4194301, F2, 15, 12) (dual of [(4194301, 15), 62914272, 13]-NRT-code), using
- digital (0, 6, 3)-net over F2, using
(249−12, 249, large)-Net over F2 — Digital
Digital (237, 249, large)-net over F2, using
- t-expansion [i] based on digital (235, 249, large)-net over F2, using
- 1 times m-reduction [i] based on digital (235, 250, large)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, large, F2, 2, 15), using
- 10 times NRT-code embedding in larger space [i] based on linear OOA(2230, 8388602, F2, 2, 15) (dual of [(8388602, 2), 16776974, 16]-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(2162, 4194301, F2, 2, 15) (dual of [(4194301, 2), 8388440, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2162, 8388602, F2, 15) (dual of [8388602, 8388440, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- OOA 2-folding [i] based on linear OA(2162, 8388602, F2, 15) (dual of [8388602, 8388440, 16]-code), using
- (u, u+v)-construction [i] based on
- 10 times NRT-code embedding in larger space [i] based on linear OOA(2230, 8388602, F2, 2, 15) (dual of [(8388602, 2), 16776974, 16]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, large, F2, 2, 15), using
- 1 times m-reduction [i] based on digital (235, 250, large)-net over F2, using
(249−12, 249, large)-Net in Base 2 — Upper bound on s
There is no (237, 249, large)-net in base 2, because
- 10 times m-reduction [i] would yield (237, 239, large)-net in base 2, but