MinT - Best Known (255−105, 255, s)-Nets in Base 2

Best Known (255−105, 255, s)-Nets in Base 2

(255−105, 255, 76)-Net over F₂ — Constructive and digital

Digital (150, 255, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 91, 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, 164, 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]

(255−105, 255, 106)-Net over F₂ — Digital

Digital (150, 255, 106)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−105, 255, 524)-Net in Base 2 — Upper bound on s

There is no (150, 255, 525)-net in base 2, because

1 times m-reduction [i] would yield (150, 254, 525)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 30874 132990 541171 688551 917680 172433 566484 587302 608841 896836 422660 845828 699424 > 2²⁵⁴ [i]