MinT - Best Known (156, 156+37, s)-Nets in Base 2

Best Known (156, 156+37, s)-Nets in Base 2

(156, 156+37, 267)-Net over F₂ — Constructive and digital

Digital (156, 193, 267)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (135, 172, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 43, 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]

(156, 156+37, 579)-Net over F₂ — Digital

Digital (156, 193, 579)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 156+37, 12251)-Net in Base 2 — Upper bound on s

There is no (156, 193, 12252)-net in base 2, because

1 times m-reduction [i] would yield (156, 192, 12252)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6284 317200 472119 537954 209991 559422 349119 049717 416564 933565 > 2¹⁹² [i]