MinT - Best Known (40, 40+49, s)-Nets in Base 2

Best Known (40, 40+49, s)-Nets in Base 2

(40, 40+49, 33)-Net over F₂ — Constructive and digital

Digital (40, 89, 33)-net over F₂, using

t-expansion [i] based on digital (39, 89, 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]

(40, 40+49, 89)-Net over F₂ — Upper bound on s (digital)

There is no digital (40, 89, 90)-net over F₂, because

9 times m-reduction [i] would yield digital (40, 80, 90)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁸⁰, 90, F₂, 40) (dual of [90, 10, 41]-code), but
  - residual code [i] would yield linear OA(2⁴⁰, 49, F₂, 20) (dual of [49, 9, 21]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(40, 40+49, 90)-Net in Base 2 — Upper bound on s

There is no (40, 89, 91)-net in base 2, because

3 times m-reduction [i] would yield (40, 86, 91)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁸⁶, 91, S₂, 46), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 3713 820117 856140 824697 372672 / 47 > 2⁸⁶ [i]