MinT - Best Known (242−101, 242, s)-Nets in Base 2

Best Known (242−101, 242, s)-Nets in Base 2

(242−101, 242, 70)-Net over F₂ — Constructive and digital

Digital (141, 242, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 71, 21)-net over F₂, using
  - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 171, 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]

(242−101, 242, 99)-Net over F₂ — Digital

Digital (141, 242, 99)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−101, 242, 480)-Net in Base 2 — Upper bound on s

There is no (141, 242, 481)-net in base 2, because

1 times m-reduction [i] would yield (141, 241, 481)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 825236 834368 190296 318785 534921 055549 166251 181068 087802 247677 015977 409136 > 2²⁴¹ [i]