MinT - Best Known (231−45, 231, s)-Nets in Base 2

Best Known (231−45, 231, s)-Nets in Base 2

(231−45, 231, 269)-Net over F₂ — Constructive and digital

Digital (186, 231, 269)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (159, 204, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 51, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(231−45, 231, 635)-Net over F₂ — Digital

Digital (186, 231, 635)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−45, 231, 12672)-Net in Base 2 — Upper bound on s

There is no (186, 231, 12673)-net in base 2, because

1 times m-reduction [i] would yield (186, 230, 12673)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1725 931863 779315 863946 491008 348717 173043 559698 094565 689979 956426 111808 > 2²³⁰ [i]