MinT - Best Known (246−57, 246, s)-Nets in Base 2

Best Known (246−57, 246, s)-Nets in Base 2

(246−57, 246, 204)-Net over F₂ — Constructive and digital

Digital (189, 246, 204)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 33, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (156, 213, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 71, 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]

(246−57, 246, 408)-Net over F₂ — Digital

Digital (189, 246, 408)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−57, 246, 4822)-Net in Base 2 — Upper bound on s

There is no (189, 246, 4823)-net in base 2, because

1 times m-reduction [i] would yield (189, 245, 4823)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 56 562131 923064 597739 303226 525442 792270 527714 801473 392338 672371 697477 173912 > 2²⁴⁵ [i]