MinT - Best Known (99−67, 99, s)-Nets in Base 2

Best Known (99−67, 99, s)-Nets in Base 2

(99−67, 99, 21)-Net over F₂ — Constructive and digital

Digital (32, 99, 21)-net over F₂, using

t-expansion [i] based on digital (21, 99, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(99−67, 99, 27)-Net over F₂ — Digital

Digital (32, 99, 27)-net over F₂, using

t-expansion [i] based on digital (31, 99, 27)-net over F₂, using
- net from sequence [i] based on digital (31, 26)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
    - Drinfeld modules of rank 1 [i]

(99−67, 99, 50)-Net in Base 2 — Upper bound on s

There is no (32, 99, 51)-net in base 2, because

3 times m-reduction [i] would yield (32, 96, 51)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2⁹⁶, 51, S₂, 2, 64), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 267650 600228 229401 496703 205376 / 13 > 2⁹⁶ [i]