MinT - Best Known (213−42, 213, s)-Nets in Base 2

Best Known (213−42, 213, s)-Nets in Base 2

(213−42, 213, 263)-Net over F₂ — Constructive and digital

Digital (171, 213, 263)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 3)-net over F₂, using
  - net from sequence [i] based on digital (0, 2)-sequence over F₂, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 0 and N(F) ≥ 3, using
      - the rational function field F₂(x) [i]
    - Niederreiter sequence [i]
    - Sobol sequence [i]
2. digital (150, 192, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 48, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(213−42, 213, 572)-Net over F₂ — Digital

Digital (171, 213, 572)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−42, 213, 9781)-Net in Base 2 — Upper bound on s

There is no (171, 213, 9782)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13165 956797 623070 615585 543096 473450 992848 867708 550148 771623 867349 > 2²¹³ [i]