MinT - Best Known (68−45, 68, s)-Nets in Base 2

Best Known (68−45, 68, s)-Nets in Base 2

(68−45, 68, 21)-Net over F₂ — Constructive and digital

Digital (23, 68, 21)-net over F₂, using

t-expansion [i] based on digital (21, 68, 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]

(68−45, 68, 22)-Net over F₂ — Digital

Digital (23, 68, 22)-net over F₂, using

net from sequence [i] based on digital (23, 21)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 23 and N(F) ≥ 22, using
  - Drinfeld modules of rank 1 [i]

(68−45, 68, 47)-Net in Base 2 — Upper bound on s

There is no (23, 68, 48)-net in base 2, because

1 times m-reduction [i] would yield (23, 67, 48)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 171 407450 933183 114258 > 2⁶⁷ [i]