MinT - Best Known (84, 169, s)-Nets in Base 2

Best Known (84, 169, s)-Nets in Base 2

Digital (84, 169, 51)-net over F₂, using

t-expansion [i] based on digital (80, 169, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 2 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

Digital (84, 169, 57)-net over F₂, using

t-expansion [i] based on digital (83, 169, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

There is no digital (84, 169, 180)-net over F₂, because

1 times m-reduction [i] would yield digital (84, 168, 180)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁸, 180, F₂, 84) (dual of [180, 12, 85]-code), but
  - residual code [i] would yield linear OA(2⁸⁴, 95, F₂, 42) (dual of [95, 11, 43]-code), but
    - adding a parity check bit [i] would yield linear OA(2⁸⁵, 96, F₂, 43) (dual of [96, 11, 44]-code), but
      - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

There is no (84, 169, 208)-net in base 2, because

1 times m-reduction [i] would yield (84, 168, 208)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 393 520451 869462 700436 624567 826565 744117 014287 558510 > 2¹⁶⁸ [i]