MinT - Best Known (171−127, 171, s)-Nets in Base 2

Best Known (171−127, 171, s)-Nets in Base 2

(171−127, 171, 33)-Net over F₂ — Constructive and digital

Digital (44, 171, 33)-net over F₂, using

t-expansion [i] based on digital (39, 171, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(171−127, 171, 34)-Net over F₂ — Digital

Digital (44, 171, 34)-net over F₂, using

t-expansion [i] based on digital (43, 171, 34)-net over F₂, using
- net from sequence [i] based on digital (43, 33)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 43 and N(F) ≥ 34, using
    - a curve from the manYPoints database [i]

(171−127, 171, 58)-Net in Base 2 — Upper bound on s

There is no (44, 171, 59)-net in base 2, because

3 times m-reduction [i] would yield (44, 168, 59)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁶⁸, 59, S₂, 3, 124), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 5986 310706 507378 352962 293074 805895 248510 699696 029696 / 15 > 2¹⁶⁸ [i]