MinT - Best Known (65−7, 65, s)-Nets in Base 2

Best Known (65−7, 65, s)-Nets in Base 2

(65−7, 65, 2097150)-Net over F₂ — Constructive and digital

Digital (58, 65, 2097150)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁶⁵, 2097150, F₂, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(2⁶⁵, 2097151, F₂, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
  - OOAs constructed from BCH codes [i]

(65−7, 65, 2097151)-Net over F₂ — Digital

Digital (58, 65, 2097151)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁶⁵, 2097151, F₂, 7, 7) (dual of [(2097151, 7), 14679992, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁶⁵, 2097151, F₂, 6, 7) (dual of [(2097151, 6), 12582841, 8]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶⁵, 2097151, F₂, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
    - OOAs constructed from BCH codes [i]

(65−7, 65, 4801275)-Net in Base 2 — Upper bound on s

There is no (58, 65, 4801276)-net in base 2, because

1 times m-reduction [i] would yield (58, 64, 4801276)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 18 446749 532494 321283 > 2⁶⁴ [i]