MinT - Best Known (137, 137+107, s)-Nets in Base 2

Best Known (137, 137+107, s)-Nets in Base 2

(137, 137+107, 67)-Net over F₂ — Constructive and digital

Digital (137, 244, 67)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 92, 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 (45, 152, 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]

(137, 137+107, 89)-Net over F₂ — Digital

Digital (137, 244, 89)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+107, 420)-Net in Base 2 — Upper bound on s

There is no (137, 244, 421)-net in base 2, because

1 times m-reduction [i] would yield (137, 243, 421)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14 743442 331225 509403 922162 619933 797872 848722 906282 200338 693579 007222 394864 > 2²⁴³ [i]