MinT - Best Known (231−44, 231, s)-Nets in Base 2

Best Known (231−44, 231, s)-Nets in Base 2

(231−44, 231, 272)-Net over F₂ — Constructive and digital

Digital (187, 231, 272)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 31, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (156, 200, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 50, 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]

(231−44, 231, 682)-Net over F₂ — Digital

Digital (187, 231, 682)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−44, 231, 13079)-Net in Base 2 — Upper bound on s

There is no (187, 231, 13080)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3453 724421 609792 890075 292096 334896 638746 271296 890317 168798 754167 423111 > 2²³¹ [i]