MinT - Best Known (231−89, 231, s)-Nets in Base 2

Best Known (231−89, 231, s)-Nets in Base 2

(231−89, 231, 76)-Net over F₂ — Constructive and digital

Digital (142, 231, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 83, 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]
2. digital (59, 148, 43)-net over F₂, using
  - net from sequence [i] based on digital (59, 42)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(231−89, 231, 114)-Net over F₂ — Digital

Digital (142, 231, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−89, 231, 583)-Net in Base 2 — Upper bound on s

There is no (142, 231, 584)-net in base 2, because

1 times m-reduction [i] would yield (142, 230, 584)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1753 937212 887601 817423 541069 016819 911942 908721 189889 420178 786009 588040 > 2²³⁰ [i]