MinT - Best Known (88−27, 88, s)-Nets in Base 2

Best Known (88−27, 88, s)-Nets in Base 2

(88−27, 88, 66)-Net over F₂ — Constructive and digital

Digital (61, 88, 66)-net over F₂, using

4 times m-reduction [i] based on digital (61, 92, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 46, 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]

(88−27, 88, 89)-Net over F₂ — Digital

Digital (61, 88, 89)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−27, 88, 567)-Net in Base 2 — Upper bound on s

There is no (61, 88, 568)-net in base 2, because

1 times m-reduction [i] would yield (61, 87, 568)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 156 815483 207284 452199 259162 > 2⁸⁷ [i]