MinT - Best Known (238−93, 238, s)-Nets in Base 2

Best Known (238−93, 238, s)-Nets in Base 2

(238−93, 238, 76)-Net over F₂ — Constructive and digital

Digital (145, 238, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (45, 91, 34)-net over F₂, using
  - net from sequence [i] based on digital (45, 33)-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 1 place 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]
2. digital (54, 147, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(238−93, 238, 113)-Net over F₂ — Digital

Digital (145, 238, 113)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−93, 238, 574)-Net in Base 2 — Upper bound on s

There is no (145, 238, 575)-net in base 2, because

1 times m-reduction [i] would yield (145, 237, 575)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 225949 364621 045091 442353 401007 548056 033289 529639 981614 573299 592798 394849 > 2²³⁷ [i]