Best Known (232−21, 232, s)-Nets in Base 2
(232−21, 232, 838860)-Net over F2 — Constructive and digital
Digital (211, 232, 838860)-net over F2, using
- 21 times duplication [i] based on digital (210, 231, 838860)-net over F2, using
- net defined by OOA [i] based on linear OOA(2231, 838860, F2, 21, 21) (dual of [(838860, 21), 17615829, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2231, 8388601, F2, 21) (dual of [8388601, 8388370, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, large, F2, 21) (dual of [large, large−231, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2231, large, F2, 21) (dual of [large, large−231, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2231, 8388601, F2, 21) (dual of [8388601, 8388370, 22]-code), using
- net defined by OOA [i] based on linear OOA(2231, 838860, F2, 21, 21) (dual of [(838860, 21), 17615829, 22]-NRT-code), using
(232−21, 232, 1048575)-Net over F2 — Digital
Digital (211, 232, 1048575)-net over F2, using
- 21 times duplication [i] based on digital (210, 231, 1048575)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 1048575, F2, 8, 21) (dual of [(1048575, 8), 8388369, 22]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2231, 8388600, F2, 21) (dual of [8388600, 8388369, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, large, F2, 21) (dual of [large, large−231, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2231, large, F2, 21) (dual of [large, large−231, 22]-code), using
- OOA 8-folding [i] based on linear OA(2231, 8388600, F2, 21) (dual of [8388600, 8388369, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 1048575, F2, 8, 21) (dual of [(1048575, 8), 8388369, 22]-NRT-code), using
(232−21, 232, large)-Net in Base 2 — Upper bound on s
There is no (211, 232, large)-net in base 2, because
- 19 times m-reduction [i] would yield (211, 213, large)-net in base 2, but