Best Known (231, 231+20, s)-Nets in Base 2
(231, 231+20, 838883)-Net over F2 — Constructive and digital
Digital (231, 251, 838883)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (11, 21, 23)-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
(231, 231+20, 1398123)-Net over F2 — Digital
Digital (231, 251, 1398123)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2251, 1398123, F2, 6, 20) (dual of [(1398123, 6), 8388487, 21]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(221, 23, F2, 6, 10) (dual of [(23, 6), 117, 11]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 21, 23)-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(221, 23, F2, 6, 10) (dual of [(23, 6), 117, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(231, 231+20, large)-Net in Base 2 — Upper bound on s
There is no (231, 251, large)-net in base 2, because
- 18 times m-reduction [i] would yield (231, 233, large)-net in base 2, but