MinT - Best Known (258−50, 258, s)-Nets in Base 2

Best Known (258−50, 258, s)-Nets in Base 2

(258−50, 258, 272)-Net over F₂ — Constructive and digital

Digital (208, 258, 272)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 34, 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 (174, 224, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 56, 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]

(258−50, 258, 707)-Net over F₂ — Digital

Digital (208, 258, 707)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−50, 258, 12972)-Net in Base 2 — Upper bound on s

There is no (208, 258, 12973)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 463710 198360 214077 098149 797730 833440 042365 192439 974993 294055 332839 436385 534420 > 2²⁵⁸ [i]