MinT - Best Known (244−103, 244, s)-Nets in Base 2

Best Known (244−103, 244, s)-Nets in Base 2

(244−103, 244, 69)-Net over F₂ — Constructive and digital

Digital (141, 244, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 90, 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, 154, 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]

(244−103, 244, 97)-Net over F₂ — Digital

Digital (141, 244, 97)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−103, 244, 468)-Net in Base 2 — Upper bound on s

There is no (141, 244, 469)-net in base 2, because

1 times m-reduction [i] would yield (141, 243, 469)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 15 140172 756982 320594 136035 592987 213571 931773 573872 666720 859873 231638 537280 > 2²⁴³ [i]