MinT - Best Known (173−87, 173, s)-Nets in Base 2

Best Known (173−87, 173, s)-Nets in Base 2

Digital (86, 173, 52)-net over F₂, using

t-expansion [i] based on digital (85, 173, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-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 3 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 (86, 173, 57)-net over F₂, using

t-expansion [i] based on digital (83, 173, 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 (86, 173, 187)-net over F₂, because

1 times m-reduction [i] would yield digital (86, 172, 187)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷², 187, F₂, 86) (dual of [187, 15, 87]-code), but
  - residual code [i] would yield linear OA(2⁸⁶, 100, F₂, 43) (dual of [100, 14, 44]-code), but
    - 1 times truncation [i] would yield linear OA(2⁸⁵, 99, F₂, 42) (dual of [99, 14, 43]-code), but
      - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

There is no (86, 173, 213)-net in base 2, because

1 times m-reduction [i] would yield (86, 172, 213)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6720 091165 515660 008361 204573 055477 378271 372346 654720 > 2¹⁷² [i]