MinT - Best Known (167−124, 167, s)-Nets in Base 2

Best Known (167−124, 167, s)-Nets in Base 2

(167−124, 167, 33)-Net over F₂ — Constructive and digital

Digital (43, 167, 33)-net over F₂, using

t-expansion [i] based on digital (39, 167, 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]

(167−124, 167, 34)-Net over F₂ — Digital

Digital (43, 167, 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]

(167−124, 167, 57)-Net in Base 2 — Upper bound on s

There is no (43, 167, 58)-net in base 2, because

2 times m-reduction [i] would yield (43, 165, 58)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁶⁵, 58, S₂, 3, 122), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 7108 743963 977511 794142 723026 332000 607606 455889 035264 / 123 > 2¹⁶⁵ [i]