MinT - Best Known (167−53, 167, s)-Nets in Base 2

Best Known (167−53, 167, s)-Nets in Base 2

(167−53, 167, 75)-Net over F₂ — Constructive and digital

Digital (114, 167, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 31, 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 (83, 136, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 68, 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]

(167−53, 167, 86)-Net in Base 2 — Constructive

(114, 167, 86)-net in base 2, using

1 times m-reduction [i] based on (114, 168, 86)-net in base 2, using
- trace code for nets [i] based on (30, 84, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [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]

(167−53, 167, 135)-Net over F₂ — Digital

Digital (114, 167, 135)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−53, 167, 843)-Net in Base 2 — Upper bound on s

There is no (114, 167, 844)-net in base 2, because

1 times m-reduction [i] would yield (114, 166, 844)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 94 051404 549017 246388 030180 084389 993535 643703 686300 > 2¹⁶⁶ [i]