MinT - Best Known (78−51, 78, s)-Nets in Base 2

Best Known (78−51, 78, s)-Nets in Base 2

(78−51, 78, 21)-Net over F₂ — Constructive and digital

Digital (27, 78, 21)-net over F₂, using

t-expansion [i] based on digital (21, 78, 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]

(78−51, 78, 24)-Net over F₂ — Digital

Digital (27, 78, 24)-net over F₂, using

t-expansion [i] based on digital (25, 78, 24)-net over F₂, using
- net from sequence [i] based on digital (25, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
    - Drinfeld modules of rank 1 [i]

(78−51, 78, 55)-Net in Base 2 — Upper bound on s

There is no (27, 78, 56)-net in base 2, because

1 times m-reduction [i] would yield (27, 77, 56)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 205213 162889 742336 137124 > 2⁷⁷ [i]