MinT - Best Known (99−55, 99, s)-Nets in Base 2

Best Known (99−55, 99, s)-Nets in Base 2

(99−55, 99, 33)-Net over F₂ — Constructive and digital

Digital (44, 99, 33)-net over F₂, using

t-expansion [i] based on digital (39, 99, 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]

(99−55, 99, 34)-Net over F₂ — Digital

Digital (44, 99, 34)-net over F₂, using

t-expansion [i] based on digital (43, 99, 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]

(99−55, 99, 96)-Net over F₂ — Upper bound on s (digital)

There is no digital (44, 99, 97)-net over F₂, because

7 times m-reduction [i] would yield digital (44, 92, 97)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁹², 97, F₂, 48) (dual of [97, 5, 49]-code), but
  - Griesmer bound [i]

(99−55, 99, 98)-Net in Base 2 — Upper bound on s

There is no (44, 99, 99)-net in base 2, because

5 times m-reduction [i] would yield (44, 94, 99)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁹⁴, 99, S₂, 50), but
  - adding a parity check bit [i] would yield OA(2⁹⁵, 100, S₂, 51), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 633825 300114 114700 748351 602688 / 13 > 2⁹⁵ [i]