MinT - Best Known (228−93, 228, s)-Nets in Base 2

Best Known (228−93, 228, s)-Nets in Base 2

(228−93, 228, 69)-Net over F₂ — Constructive and digital

Digital (135, 228, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 65, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 163, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(228−93, 228, 99)-Net over F₂ — Digital

Digital (135, 228, 99)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−93, 228, 485)-Net in Base 2 — Upper bound on s

There is no (135, 228, 486)-net in base 2, because

1 times m-reduction [i] would yield (135, 227, 486)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 222 292956 271858 142667 670917 463335 512713 299424 755646 887701 642237 894368 > 2²²⁷ [i]