Best Known (86, 86+8, s)-Nets in Base 2
(86, 86+8, 2097150)-Net over F2 — Constructive and digital
Digital (86, 94, 2097150)-net over F2, using
- 21 times duplication [i] based on digital (85, 93, 2097150)-net over F2, using
- t-expansion [i] based on digital (84, 93, 2097150)-net over F2, using
- net defined by OOA [i] based on linear OOA(293, 2097150, F2, 9, 9) (dual of [(2097150, 9), 18874257, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(293, 2097150, F2, 8, 9) (dual of [(2097150, 8), 16777107, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(293, 8388601, F2, 9) (dual of [8388601, 8388508, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(293, large, F2, 9) (dual of [large, large−93, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(293, large, F2, 9) (dual of [large, large−93, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(293, 8388601, F2, 9) (dual of [8388601, 8388508, 10]-code), using
- appending kth column [i] based on linear OOA(293, 2097150, F2, 8, 9) (dual of [(2097150, 8), 16777107, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(293, 2097150, F2, 9, 9) (dual of [(2097150, 9), 18874257, 10]-NRT-code), using
- t-expansion [i] based on digital (84, 93, 2097150)-net over F2, using
(86, 86+8, 2796201)-Net over F2 — Digital
Digital (86, 94, 2796201)-net over F2, using
- 22 times duplication [i] based on digital (84, 92, 2796201)-net over F2, using
- net defined by OOA [i] based on linear OOA(292, 2796201, F2, 8, 8) (dual of [(2796201, 8), 22369516, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(292, 2796201, F2, 7, 8) (dual of [(2796201, 7), 19573315, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(292, 2796201, F2, 3, 8) (dual of [(2796201, 3), 8388511, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(292, large, F2, 8) (dual of [large, large−92, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 3-folding [i] based on linear OA(292, large, F2, 8) (dual of [large, large−92, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(292, 2796201, F2, 3, 8) (dual of [(2796201, 3), 8388511, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(292, 2796201, F2, 7, 8) (dual of [(2796201, 7), 19573315, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(292, 2796201, F2, 8, 8) (dual of [(2796201, 8), 22369516, 9]-NRT-code), using
(86, 86+8, large)-Net in Base 2 — Upper bound on s
There is no (86, 94, large)-net in base 2, because
- 6 times m-reduction [i] would yield (86, 88, large)-net in base 2, but