MinT - Best Known (217−49, 217, s)-Nets in Base 2

Best Known (217−49, 217, s)-Nets in Base 2

(217−49, 217, 203)-Net over F₂ — Constructive and digital

Digital (168, 217, 203)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 8)-net over F₂, using
  - net from sequence [i] based on digital (4, 7)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 4 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (140, 189, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 63, 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]

(217−49, 217, 392)-Net over F₂ — Digital

Digital (168, 217, 392)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−49, 217, 4983)-Net in Base 2 — Upper bound on s

There is no (168, 217, 4984)-net in base 2, because

1 times m-reduction [i] would yield (168, 216, 4984)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 105416 369040 981132 839371 876181 121046 137749 722028 003201 366562 296876 > 2²¹⁶ [i]