MinT - Best Known (63, 63+43, s)-Nets in Base 3

Best Known (63, 63+43, s)-Nets in Base 3

(63, 63+43, 80)-Net over F₃ — Constructive and digital

Digital (63, 106, 80)-net over F₃, using

4 times m-reduction [i] based on digital (63, 110, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 55, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(63, 63+43, 108)-Net over F₃ — Digital

Digital (63, 106, 108)-net over F₃, using

trace code for nets [i] based on digital (10, 53, 54)-net over F₉, using
- net from sequence [i] based on digital (10, 53)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 10 and N(F) ≥ 54, using
    - a curve from the manYPoints database [i]

(63, 63+43, 1034)-Net in Base 3 — Upper bound on s

There is no (63, 106, 1035)-net in base 3, because

1 times m-reduction [i] would yield (63, 105, 1035)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 127 199214 542653 888439 504310 609044 574004 693976 429487 > 3¹⁰⁵ [i]