MinT - Best Known (229−45, 229, s)-Nets in Base 2

Best Known (229−45, 229, s)-Nets in Base 2

(229−45, 229, 267)-Net over F₂ — Constructive and digital

Digital (184, 229, 267)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (159, 204, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 51, 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]

(229−45, 229, 613)-Net over F₂ — Digital

Digital (184, 229, 613)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−45, 229, 11896)-Net in Base 2 — Upper bound on s

There is no (184, 229, 11897)-net in base 2, because

1 times m-reduction [i] would yield (184, 228, 11897)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 431 410194 932758 659355 732806 899193 950249 546855 002763 794639 896757 270548 > 2²²⁸ [i]