MinT - Best Known (72, 72+37, s)-Nets in Base 2

Best Known (72, 72+37, s)-Nets in Base 2

(72, 72+37, 66)-Net over F₂ — Constructive and digital

Digital (72, 109, 66)-net over F₂, using

5 times m-reduction [i] based on digital (72, 114, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 57, 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]

(72, 72+37, 83)-Net over F₂ — Digital

Digital (72, 109, 83)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+37, 457)-Net in Base 2 — Upper bound on s

There is no (72, 109, 458)-net in base 2, because

1 times m-reduction [i] would yield (72, 108, 458)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 330 071702 110825 687856 228845 397680 > 2¹⁰⁸ [i]