MinT - Best Known (245−47, 245, s)-Nets in Base 2

Best Known (245−47, 245, s)-Nets in Base 2

(245−47, 245, 272)-Net over F₂ — Constructive and digital

Digital (198, 245, 272)-net over F₂, using

2¹ times duplication [i] based on digital (197, 244, 272)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 32, 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]
    - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
  2. digital (165, 212, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 53, 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]

(245−47, 245, 700)-Net over F₂ — Digital

Digital (198, 245, 700)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−47, 245, 14688)-Net in Base 2 — Upper bound on s

There is no (198, 245, 14689)-net in base 2, because

1 times m-reduction [i] would yield (198, 244, 14689)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28 274598 848724 657289 177026 104750 232460 256386 548070 092270 554638 030379 336192 > 2²⁴⁴ [i]