Best Known (229, 229+20, s)-Nets in Base 2
(229, 229+20, 838877)-Net over F2 — Constructive and digital
Digital (229, 249, 838877)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (9, 19, 17)-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
(229, 229+20, 1398117)-Net over F2 — Digital
Digital (229, 249, 1398117)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2249, 1398117, F2, 6, 20) (dual of [(1398117, 6), 8388453, 21]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(219, 17, F2, 6, 10) (dual of [(17, 6), 83, 11]-NRT-code), using
- extracting embedded OOA [i] based on digital (9, 19, 17)-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(219, 17, F2, 6, 10) (dual of [(17, 6), 83, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(229, 229+20, large)-Net in Base 2 — Upper bound on s
There is no (229, 249, large)-net in base 2, because
- 18 times m-reduction [i] would yield (229, 231, large)-net in base 2, but