MinT - Best Known (27, 27+7, s)-Nets in Base 2

Best Known (27, 27+7, s)-Nets in Base 2

Digital (27, 34, 1022)-net over F₂, using

2² times duplication [i] based on digital (25, 32, 1022)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2³², 1022, F₂, 7, 7) (dual of [(1022, 7), 7122, 8]-NRT-code), using
  - OOA stacking with additional row [i] based on linear OOA(2³², 1023, F₂, 3, 7) (dual of [(1023, 3), 3037, 8]-NRT-code), using
    - OOAs constructed from BCH codes [i]

Digital (27, 34, 1023)-net over F₂, using

2² times duplication [i] based on digital (25, 32, 1023)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2³², 1023, F₂, 7, 7) (dual of [(1023, 7), 7129, 8]-NRT-code), using
  - appending kth column [i] based on linear OOA(2³², 1023, F₂, 6, 7) (dual of [(1023, 6), 6106, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³², 1023, F₂, 3, 7) (dual of [(1023, 3), 3037, 8]-NRT-code), using
      - OOAs constructed from BCH codes [i]

There is no (27, 34, 3718)-net in base 2, because

1 times m-reduction [i] would yield (27, 33, 3718)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 8593 642058 > 2³³ [i]