Best Known (56, 56+9, s)-Nets in Base 2
(56, 56+9, 16384)-Net over F2 — Constructive and digital
Digital (56, 65, 16384)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 16384, F2, 9, 9) (dual of [(16384, 9), 147391, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(265, 16384, F2, 8, 9) (dual of [(16384, 8), 131007, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(265, 65537, F2, 9) (dual of [65537, 65472, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, 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(265, 65537, F2, 9) (dual of [65537, 65472, 10]-code), using
- appending kth column [i] based on linear OOA(265, 16384, F2, 8, 9) (dual of [(16384, 8), 131007, 10]-NRT-code), using
(56, 56+9, 145049)-Net in Base 2 — Upper bound on s
There is no (56, 65, 145050)-net in base 2, because
- 1 times m-reduction [i] would yield (56, 64, 145050)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 18 446991 973339 112826 > 264 [i]