MinT - Best Known (213−83, 213, s)-Nets in Base 2

Best Known (213−83, 213, s)-Nets in Base 2

(213−83, 213, 69)-Net over F₂ — Constructive and digital

Digital (130, 213, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 60, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 153, 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]

(213−83, 213, 103)-Net over F₂ — Digital

Digital (130, 213, 103)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−83, 213, 523)-Net in Base 2 — Upper bound on s

There is no (130, 213, 524)-net in base 2, because

1 times m-reduction [i] would yield (130, 212, 524)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6991 373378 899064 240817 072671 901601 854463 457415 457278 807180 569520 > 2²¹² [i]