Best Known (212, 212+20, s)-Nets in Base 2
(212, 212+20, 838860)-Net over F2 — Constructive and digital
Digital (212, 232, 838860)-net over F2, using
- 21 times duplication [i] based on digital (211, 231, 838860)-net over F2, using
- t-expansion [i] based on 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
- t-expansion [i] based on digital (210, 231, 838860)-net over F2, using
(212, 212+20, 1198371)-Net over F2 — Digital
Digital (212, 232, 1198371)-net over F2, using
- 22 times duplication [i] based on digital (210, 230, 1198371)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2230, 1198371, F2, 7, 20) (dual of [(1198371, 7), 8388367, 21]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2230, 8388597, F2, 20) (dual of [8388597, 8388367, 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 7-folding [i] based on linear OA(2230, 8388597, F2, 20) (dual of [8388597, 8388367, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2230, 1198371, F2, 7, 20) (dual of [(1198371, 7), 8388367, 21]-NRT-code), using
(212, 212+20, large)-Net in Base 2 — Upper bound on s
There is no (212, 232, large)-net in base 2, because
- 18 times m-reduction [i] would yield (212, 214, large)-net in base 2, but