Best Known (68−8, 68, s)-Nets in Base 2
(68−8, 68, 32768)-Net over F2 — Constructive and digital
Digital (60, 68, 32768)-net over F2, using
- net defined by OOA [i] based on linear OOA(268, 32768, F2, 8, 8) (dual of [(32768, 8), 262076, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- 1 times truncation [i] based on linear OA(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, 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(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
(68−8, 68, 43690)-Net over F2 — Digital
Digital (60, 68, 43690)-net over F2, using
- net defined by OOA [i] based on linear OOA(268, 43690, F2, 8, 8) (dual of [(43690, 8), 349452, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(268, 43690, F2, 7, 8) (dual of [(43690, 7), 305762, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(268, 43690, F2, 3, 8) (dual of [(43690, 3), 131002, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(268, 131070, F2, 8) (dual of [131070, 131002, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- 1 times truncation [i] based on linear OA(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, 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(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- OOA 3-folding [i] based on linear OA(268, 131070, F2, 8) (dual of [131070, 131002, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(268, 43690, F2, 3, 8) (dual of [(43690, 3), 131002, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(268, 43690, F2, 7, 8) (dual of [(43690, 7), 305762, 9]-NRT-code), using
(68−8, 68, 290104)-Net in Base 2 — Upper bound on s
There is no (60, 68, 290105)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 295 149837 358631 234346 > 268 [i]