MinT - Best Known (257−49, 257, s)-Nets in Base 2

Best Known (257−49, 257, s)-Nets in Base 2

(257−49, 257, 275)-Net over F₂ — Constructive and digital

Digital (208, 257, 275)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (13, 37, 15)-net over F₂, using
  - net from sequence [i] based on digital (13, 14)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 13 and N(F) ≥ 15, 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]

(257−49, 257, 743)-Net over F₂ — Digital

Digital (208, 257, 743)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−49, 257, 15899)-Net in Base 2 — Upper bound on s

There is no (208, 257, 15900)-net in base 2, because

1 times m-reduction [i] would yield (208, 256, 15900)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 115890 381453 505108 514311 225847 812517 168643 557554 287739 872289 450830 397329 644621 > 2²⁵⁶ [i]