MinT - Best Known (226−91, 226, s)-Nets in Base 2

Best Known (226−91, 226, s)-Nets in Base 2

(226−91, 226, 69)-Net over F₂ — Constructive and digital

Digital (135, 226, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 84, 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 (51, 142, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(226−91, 226, 101)-Net over F₂ — Digital

Digital (135, 226, 101)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−91, 226, 500)-Net in Base 2 — Upper bound on s

There is no (135, 226, 501)-net in base 2, because

1 times m-reduction [i] would yield (135, 225, 501)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 56 222571 483184 458702 391330 170680 902456 344329 926408 938969 302591 472052 > 2²²⁵ [i]