Best Known (68, 77, s)-Nets in Base 2
(68, 77, 131072)-Net over F2 — Constructive and digital
Digital (68, 77, 131072)-net over F2, using
- net defined by OOA [i] based on linear OOA(277, 131072, F2, 9, 9) (dual of [(131072, 9), 1179571, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(277, 131072, F2, 8, 9) (dual of [(131072, 8), 1048499, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(277, 524289, F2, 9) (dual of [524289, 524212, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, 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(277, 524289, F2, 9) (dual of [524289, 524212, 10]-code), using
- appending kth column [i] based on linear OOA(277, 131072, F2, 8, 9) (dual of [(131072, 8), 1048499, 10]-NRT-code), using
(68, 77, 1160434)-Net in Base 2 — Upper bound on s
There is no (68, 77, 1160435)-net in base 2, because
- 1 times m-reduction [i] would yield (68, 76, 1160435)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 75557 967736 833270 500916 > 276 [i]