Best Known (258−21, 258, s)-Nets in Base 2
(258−21, 258, 838890)-Net over F2 — Constructive and digital
Digital (237, 258, 838890)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (17, 27, 30)-net over F2, using
- 3 times m-reduction [i] based on digital (17, 30, 30)-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 (17, 27, 30)-net over F2, using
(258−21, 258, 1398131)-Net over F2 — Digital
Digital (237, 258, 1398131)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 1398131, F2, 6, 21) (dual of [(1398131, 6), 8388528, 22]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(227, 31, F2, 6, 10) (dual of [(31, 6), 159, 11]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(227, 31, F2, 2, 10) (dual of [(31, 2), 35, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(227, 62, F2, 10) (dual of [62, 35, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(227, 63, F2, 10) (dual of [63, 36, 11]-code), using
- the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(227, 63, F2, 10) (dual of [63, 36, 11]-code), using
- OOA 2-folding [i] based on linear OA(227, 62, F2, 10) (dual of [62, 35, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(227, 31, F2, 2, 10) (dual of [(31, 2), 35, 11]-NRT-code), 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(227, 31, F2, 6, 10) (dual of [(31, 6), 159, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(258−21, 258, large)-Net in Base 2 — Upper bound on s
There is no (237, 258, large)-net in base 2, because
- 19 times m-reduction [i] would yield (237, 239, large)-net in base 2, but