MinT - Best Known (180−73, 180, s)-Nets in Base 2

Best Known (180−73, 180, s)-Nets in Base 2

(180−73, 180, 66)-Net over F₂ — Constructive and digital

Digital (107, 180, 66)-net over F₂, using

4 times m-reduction [i] based on digital (107, 184, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 92, 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]

(180−73, 180, 82)-Net over F₂ — Digital

Digital (107, 180, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−73, 180, 397)-Net in Base 2 — Upper bound on s

There is no (107, 180, 398)-net in base 2, because

1 times m-reduction [i] would yield (107, 179, 398)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 798086 134272 600359 546333 576227 210435 974726 772055 812108 > 2¹⁷⁹ [i]