Best Known (48, 57, s)-Nets in Base 2
(48, 57, 4096)-Net over F2 — Constructive and digital
Digital (48, 57, 4096)-net over F2, using
- net defined by OOA [i] based on linear OOA(257, 4096, F2, 9, 9) (dual of [(4096, 9), 36807, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(257, 4096, F2, 8, 9) (dual of [(4096, 8), 32711, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- appending kth column [i] based on linear OOA(257, 4096, F2, 8, 9) (dual of [(4096, 8), 32711, 10]-NRT-code), using
(48, 57, 4638)-Net over F2 — Digital
Digital (48, 57, 4638)-net over F2, using
- net defined by OOA [i] based on linear OOA(257, 4638, F2, 9, 9) (dual of [(4638, 9), 41685, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(257, 4638, F2, 8, 9) (dual of [(4638, 8), 37047, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(257, 4638, F2, 3, 9) (dual of [(4638, 3), 13857, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(257, 5461, F2, 3, 9) (dual of [(5461, 3), 16326, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(257, 16383, F2, 9) (dual of [16383, 16326, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using
- OOA 3-folding [i] based on linear OA(257, 16383, F2, 9) (dual of [16383, 16326, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(257, 5461, F2, 3, 9) (dual of [(5461, 3), 16326, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(257, 4638, F2, 3, 9) (dual of [(4638, 3), 13857, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(257, 4638, F2, 8, 9) (dual of [(4638, 8), 37047, 10]-NRT-code), using
(48, 57, 36258)-Net in Base 2 — Upper bound on s
There is no (48, 57, 36259)-net in base 2, because
- 1 times m-reduction [i] would yield (48, 56, 36259)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 72063 527584 777712 > 256 [i]