MinT - Best Known (257−61, 257, s)-Nets in Base 2

Best Known (257−61, 257, s)-Nets in Base 2

(257−61, 257, 201)-Net over F₂ — Constructive and digital

Digital (196, 257, 201)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 6)-net over F₂, using
  - net from sequence [i] based on digital (2, 5)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (164, 225, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 75, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(257−61, 257, 397)-Net over F₂ — Digital

Digital (196, 257, 397)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−61, 257, 4418)-Net in Base 2 — Upper bound on s

There is no (196, 257, 4419)-net in base 2, because

1 times m-reduction [i] would yield (196, 256, 4419)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 116416 930969 986772 568842 134442 498588 013107 434473 585098 304370 603161 639233 311600 > 2²⁵⁶ [i]