Best Known (68, 68+4, s)-Nets in Base 8
(68, 68+4, large)-Net over F8 — Constructive and digital
Digital (68, 72, large)-net over F8, using
- 88 times duplication [i] based on digital (60, 64, large)-net over F8, using
- t-expansion [i] based on digital (57, 64, large)-net over F8, using
- trace code for nets [i] based on digital (25, 32, 5592401)-net over F64, using
- net defined by OOA [i] based on linear OOA(6432, 5592401, F64, 9, 7) (dual of [(5592401, 9), 50331577, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(6432, 5592402, F64, 3, 7) (dual of [(5592402, 3), 16777174, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(647, large, F64, 3, 3), using
- appending kth column [i] based on linear OOA(647, large, F64, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(647, large, F64, 3) (dual of [large, large−7, 4]-code), using
- appending kth column [i] based on linear OOA(647, large, F64, 2, 3), using
- linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- linear OOA(647, large, F64, 3, 3), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(6432, 5592402, F64, 3, 7) (dual of [(5592402, 3), 16777174, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6432, 5592401, F64, 9, 7) (dual of [(5592401, 9), 50331577, 8]-NRT-code), using
- trace code for nets [i] based on digital (25, 32, 5592401)-net over F64, using
- t-expansion [i] based on digital (57, 64, large)-net over F8, using
(68, 68+4, large)-Net in Base 8 — Upper bound on s
There is no (68, 72, large)-net in base 8, because
- 2 times m-reduction [i] would yield (68, 70, large)-net in base 8, but