Best Known (48, 48+8, s)-Nets in Base 2
(48, 48+8, 4096)-Net over F2 — Constructive and digital
Digital (48, 56, 4096)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 4096, F2, 8, 8) (dual of [(4096, 8), 32712, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
(48, 48+8, 5461)-Net over F2 — Digital
Digital (48, 56, 5461)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 5461, F2, 8, 8) (dual of [(5461, 8), 43632, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(256, 5461, F2, 7, 8) (dual of [(5461, 7), 38171, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(256, 5461, F2, 3, 8) (dual of [(5461, 3), 16327, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(256, 16383, F2, 8) (dual of [16383, 16327, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- OOA 3-folding [i] based on linear OA(256, 16383, F2, 8) (dual of [16383, 16327, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(256, 5461, F2, 3, 8) (dual of [(5461, 3), 16327, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(256, 5461, F2, 7, 8) (dual of [(5461, 7), 38171, 9]-NRT-code), using
(48, 48+8, 36258)-Net in Base 2 — Upper bound on s
There is no (48, 56, 36259)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 72063 527584 777712 > 256 [i]