Best Known (57−7, 57, s)-Nets in Base 2
(57−7, 57, 262142)-Net over F2 — Constructive and digital
Digital (50, 57, 262142)-net over F2, using
- 21 times duplication [i] based on digital (49, 56, 262142)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 262142, F2, 7, 7) (dual of [(262142, 7), 1834938, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(256, 262143, F2, 3, 7) (dual of [(262143, 3), 786373, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(256, 262142, F2, 7, 7) (dual of [(262142, 7), 1834938, 8]-NRT-code), using
(57−7, 57, 262143)-Net over F2 — Digital
Digital (50, 57, 262143)-net over F2, using
- 21 times duplication [i] based on digital (49, 56, 262143)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 262143, F2, 7, 7) (dual of [(262143, 7), 1834945, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(256, 262143, F2, 6, 7) (dual of [(262143, 6), 1572802, 8]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(256, 262143, F2, 3, 7) (dual of [(262143, 3), 786373, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(256, 262143, F2, 6, 7) (dual of [(262143, 6), 1572802, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(256, 262143, F2, 7, 7) (dual of [(262143, 7), 1834945, 8]-NRT-code), using
(57−7, 57, 756150)-Net in Base 2 — Upper bound on s
There is no (50, 57, 756151)-net in base 2, because
- 1 times m-reduction [i] would yield (50, 56, 756151)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 72057 839119 676158 > 256 [i]