MinT - Best Known (210−89, 210, s)-Nets in Base 2

Best Known (210−89, 210, s)-Nets in Base 2

(210−89, 210, 66)-Net over F₂ — Constructive and digital

Digital (121, 210, 66)-net over F₂, using

2 times m-reduction [i] based on digital (121, 212, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 106, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(210−89, 210, 84)-Net over F₂ — Digital

Digital (121, 210, 84)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−89, 210, 402)-Net in Base 2 — Upper bound on s

There is no (121, 210, 403)-net in base 2, because

1 times m-reduction [i] would yield (121, 209, 403)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 843 179684 010033 346670 335083 555149 289918 639558 719835 099276 613456 > 2²⁰⁹ [i]