MinT - Best Known (64−29, 64, s)-Nets in Base 2

Best Known (64−29, 64, s)-Nets in Base 2

(64−29, 64, 28)-Net over F₂ — Constructive and digital

Digital (35, 64, 28)-net over F₂, using

trace code for nets [i] based on digital (3, 32, 14)-net over F₄, using
- net from sequence [i] based on digital (3, 13)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
    - an explicitly constructive algebraic function field [i]

(64−29, 64, 29)-Net over F₂ — Digital

Digital (35, 64, 29)-net over F₂, using

net from sequence [i] based on digital (35, 28)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 35 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(64−29, 64, 116)-Net in Base 2 — Upper bound on s

There is no (35, 64, 117)-net in base 2, because

1 times m-reduction [i] would yield (35, 63, 117)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶³, 117, S₂, 28), but
  - the linear programming bound shows that MÂ â‰¥ 16946 634974 952621 699046 667146 230655 213597 884416 / 1729 294234 965439 779438 380995 > 2⁶³ [i]