MinT - Best Known (77−45, 77, s)-Nets in Base 2

Best Known (77−45, 77, s)-Nets in Base 2

(77−45, 77, 21)-Net over F₂ — Constructive and digital

Digital (32, 77, 21)-net over F₂, using

t-expansion [i] based on digital (21, 77, 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]

(77−45, 77, 27)-Net over F₂ — Digital

Digital (32, 77, 27)-net over F₂, using

t-expansion [i] based on digital (31, 77, 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]

(77−45, 77, 70)-Net in Base 2 — Upper bound on s

There is no (32, 77, 71)-net in base 2, because

1 times m-reduction [i] would yield (32, 76, 71)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 77347 671686 631281 635644 > 2⁷⁶ [i]