MinT - Best Known (84−41, 84, s)-Nets in Base 2

Best Known (84−41, 84, s)-Nets in Base 2

(84−41, 84, 33)-Net over F₂ — Constructive and digital

Digital (43, 84, 33)-net over F₂, using

t-expansion [i] based on digital (39, 84, 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]

(84−41, 84, 34)-Net over F₂ — Digital

Digital (43, 84, 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]

(84−41, 84, 102)-Net over F₂ — Upper bound on s (digital)

There is no digital (43, 84, 103)-net over F₂, because

1 times m-reduction [i] would yield digital (43, 83, 103)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁸³, 103, F₂, 40) (dual of [103, 20, 41]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁸⁴, 104, F₂, 41) (dual of [104, 20, 42]-code), but
    - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(84−41, 84, 103)-Net in Base 2 — Upper bound on s

There is no (43, 84, 104)-net in base 2, because

1 times m-reduction [i] would yield (43, 83, 104)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁸³, 104, S₂, 40), but
  - the linear programming bound shows that MÂ â‰¥ 455561 934457 019941 162877 714432 / 37961 > 2⁸³ [i]