Best Known (226, 226+20, s)-Nets in Base 2
(226, 226+20, 838871)-Net over F2 — Constructive and digital
Digital (226, 246, 838871)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (6, 16, 11)-net over F2, using
- digital (210, 230, 838860)-net over F2, using
- net defined by OOA [i] based on linear OOA(2230, 838860, F2, 20, 20) (dual of [(838860, 20), 16776970, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2230, 8388600, F2, 20) (dual of [8388600, 8388370, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2230, large, F2, 20) (dual of [large, large−230, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2230, large, F2, 20) (dual of [large, large−230, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2230, 8388600, F2, 20) (dual of [8388600, 8388370, 21]-code), using
- net defined by OOA [i] based on linear OOA(2230, 838860, F2, 20, 20) (dual of [(838860, 20), 16776970, 21]-NRT-code), using
(226, 226+20, 1398111)-Net over F2 — Digital
Digital (226, 246, 1398111)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2246, 1398111, F2, 6, 20) (dual of [(1398111, 6), 8388420, 21]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(216, 11, F2, 6, 10) (dual of [(11, 6), 50, 11]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 16, 11)-net over F2, using
- linear OOA(2230, 1398100, F2, 6, 20) (dual of [(1398100, 6), 8388370, 21]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2230, 8388600, F2, 20) (dual of [8388600, 8388370, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2230, large, F2, 20) (dual of [large, large−230, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2230, large, F2, 20) (dual of [large, large−230, 21]-code), using
- OOA 6-folding [i] based on linear OA(2230, 8388600, F2, 20) (dual of [8388600, 8388370, 21]-code), using
- linear OOA(216, 11, F2, 6, 10) (dual of [(11, 6), 50, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(226, 226+20, large)-Net in Base 2 — Upper bound on s
There is no (226, 246, large)-net in base 2, because
- 18 times m-reduction [i] would yield (226, 228, large)-net in base 2, but