MinT - Best Known (140−43, 140, s)-Nets in Base 2

Best Known (140−43, 140, s)-Nets in Base 2

(140−43, 140, 73)-Net over F₂ — Constructive and digital

Digital (97, 140, 73)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (73, 116, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 58, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(140−43, 140, 84)-Net in Base 2 — Constructive

(97, 140, 84)-net in base 2, using

trace code for nets [i] based on (27, 70, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(140−43, 140, 126)-Net over F₂ — Digital

Digital (97, 140, 126)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−43, 140, 822)-Net in Base 2 — Upper bound on s

There is no (97, 140, 823)-net in base 2, because

1 times m-reduction [i] would yield (97, 139, 823)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 702907 461620 697539 510732 124365 498742 079284 > 2¹³⁹ [i]