MinT - Best Known (91−47, 91, s)-Nets in Base 2

Best Known (91−47, 91, s)-Nets in Base 2

(91−47, 91, 33)-Net over F₂ — Constructive and digital

Digital (44, 91, 33)-net over F₂, using

t-expansion [i] based on digital (39, 91, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(91−47, 91, 34)-Net over F₂ — Digital

Digital (44, 91, 34)-net over F₂, using

t-expansion [i] based on digital (43, 91, 34)-net over F₂, using
- net from sequence [i] based on digital (43, 33)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 43 and N(F) ≥ 34, using
    - a curve from the manYPoints database [i]

(91−47, 91, 97)-Net over F₂ — Upper bound on s (digital)

There is no digital (44, 91, 98)-net over F₂, because

3 times m-reduction [i] would yield digital (44, 88, 98)-net over F₂, but
- extracting embedded orthogonal array [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]

(91−47, 91, 99)-Net in Base 2 — Upper bound on s

There is no (44, 91, 100)-net in base 2, because

1 times m-reduction [i] would yield (44, 90, 100)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁹⁰, 100, S₂, 46), but
  - the linear programming bound shows that MÂ â‰¥ 1 861861 819085 211933 448282 832896 / 1305 > 2⁹⁰ [i]