Best Known (235, 235+21, s)-Nets in Base 2
(235, 235+21, 838887)-Net over F2 — Constructive and digital
Digital (235, 256, 838887)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (15, 25, 27)-net over F2, using
- 2 times m-reduction [i] based on digital (15, 27, 27)-net over F2, using
- 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
- digital (15, 25, 27)-net over F2, using
(235, 235+21, 1398127)-Net over F2 — Digital
Digital (235, 256, 1398127)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 1398127, F2, 6, 21) (dual of [(1398127, 6), 8388506, 22]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(225, 27, F2, 6, 10) (dual of [(27, 6), 137, 11]-NRT-code), using
- extracting embedded OOA [i] based on digital (15, 25, 27)-net over F2, using
- 2 times m-reduction [i] based on digital (15, 27, 27)-net over F2, using
- extracting embedded OOA [i] based on digital (15, 25, 27)-net over F2, using
- linear OOA(2231, 1398100, F2, 6, 21) (dual of [(1398100, 6), 8388369, 22]-NRT-code), using
- OOA 6-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 6-folding [i] based on linear OA(2231, 8388600, F2, 21) (dual of [8388600, 8388369, 22]-code), using
- linear OOA(225, 27, F2, 6, 10) (dual of [(27, 6), 137, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(235, 235+21, large)-Net in Base 2 — Upper bound on s
There is no (235, 256, large)-net in base 2, because
- 19 times m-reduction [i] would yield (235, 237, large)-net in base 2, but