MinT - Best Known (244−105, 244, s)-Nets in Base 2

Best Known (244−105, 244, s)-Nets in Base 2

(244−105, 244, 68)-Net over F₂ — Constructive and digital

Digital (139, 244, 68)-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 (48, 153, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-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 2 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]

(244−105, 244, 93)-Net over F₂ — Digital

Digital (139, 244, 93)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−105, 244, 443)-Net in Base 2 — Upper bound on s

There is no (139, 244, 444)-net in base 2, because

1 times m-reduction [i] would yield (139, 243, 444)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 15 014157 155774 092185 034643 504367 314431 223929 593810 450282 566263 146781 634382 > 2²⁴³ [i]