Best Known (228, 228+21, s)-Nets in Base 2
(228, 228+21, 838875)-Net over F2 — Constructive and digital
Digital (228, 249, 838875)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (8, 18, 15)-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
(228, 228+21, 1198386)-Net over F2 — Digital
Digital (228, 249, 1198386)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2249, 1198386, F2, 7, 21) (dual of [(1198386, 7), 8388453, 22]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(218, 15, F2, 7, 10) (dual of [(15, 7), 87, 11]-NRT-code), using
- extracting embedded OOA [i] based on digital (8, 18, 15)-net over F2, using
- linear OOA(2231, 1198371, F2, 7, 21) (dual of [(1198371, 7), 8388366, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2231, 8388597, F2, 21) (dual of [8388597, 8388366, 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 7-folding [i] based on linear OA(2231, 8388597, F2, 21) (dual of [8388597, 8388366, 22]-code), using
- linear OOA(218, 15, F2, 7, 10) (dual of [(15, 7), 87, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(228, 228+21, large)-Net in Base 2 — Upper bound on s
There is no (228, 249, large)-net in base 2, because
- 19 times m-reduction [i] would yield (228, 230, large)-net in base 2, but