Best Known (239−20, 239, s)-Nets in Base 2
(239−20, 239, 838860)-Net over F2 — Constructive and digital
Digital (219, 239, 838860)-net over F2, using
- 28 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
(239−20, 239, 1398102)-Net over F2 — Digital
Digital (219, 239, 1398102)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 1398102, F2, 6, 20) (dual of [(1398102, 6), 8388373, 21]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2233, 1398101, F2, 6, 20) (dual of [(1398101, 6), 8388373, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2233, 2796202, F2, 3, 20) (dual of [(2796202, 3), 8388373, 21]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2230, 2796201, F2, 3, 20) (dual of [(2796201, 3), 8388373, 21]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(2230, large, F2, 20) (dual of [large, large−230, 21]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2230, 2796201, F2, 3, 20) (dual of [(2796201, 3), 8388373, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2233, 2796202, F2, 3, 20) (dual of [(2796202, 3), 8388373, 21]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2233, 1398101, F2, 6, 20) (dual of [(1398101, 6), 8388373, 21]-NRT-code), using
(239−20, 239, large)-Net in Base 2 — Upper bound on s
There is no (219, 239, large)-net in base 2, because
- 18 times m-reduction [i] would yield (219, 221, large)-net in base 2, but