MinT - Best Known (254−49, 254, s)-Nets in Base 2

Best Known (254−49, 254, s)-Nets in Base 2

(254−49, 254, 272)-Net over F₂ — Constructive and digital

Digital (205, 254, 272)-net over F₂, using

2¹ times duplication [i] based on digital (204, 253, 272)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 33, 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]
  2. digital (171, 220, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 55, 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]

(254−49, 254, 708)-Net over F₂ — Digital

Digital (205, 254, 708)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−49, 254, 14576)-Net in Base 2 — Upper bound on s

There is no (205, 254, 14577)-net in base 2, because

1 times m-reduction [i] would yield (205, 253, 14577)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14476 208690 178426 152716 879153 520323 082590 868348 713819 397187 221385 246635 384948 > 2²⁵³ [i]