MinT - Best Known (177−89, 177, s)-Nets in Base 2

Best Known (177−89, 177, s)-Nets in Base 2

Digital (88, 177, 52)-net over F₂, using

t-expansion [i] based on digital (85, 177, 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 (88, 177, 57)-net over F₂, using

t-expansion [i] based on digital (83, 177, 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 (88, 177, 187)-net over F₂, because

1 times m-reduction [i] would yield digital (88, 176, 187)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁶, 187, F₂, 88) (dual of [187, 11, 89]-code), but
  - residual code [i] would yield linear OA(2⁸⁸, 98, F₂, 44) (dual of [98, 10, 45]-code), but
    - adding a parity check bit [i] would yield linear OA(2⁸⁹, 99, F₂, 45) (dual of [99, 10, 46]-code), but
      - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

There is no (88, 177, 217)-net in base 2, because

1 times m-reduction [i] would yield (88, 176, 217)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 97592 358721 932449 530859 281398 523875 470014 469508 567800 > 2¹⁷⁶ [i]