MinT - Best Known (61, 61+41, s)-Nets in Base 3

Best Known (61, 61+41, s)-Nets in Base 3

(61, 61+41, 80)-Net over F₃ — Constructive and digital

Digital (61, 102, 80)-net over F₃, using

4 times m-reduction [i] based on digital (61, 106, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 53, 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]

(61, 61+41, 108)-Net over F₃ — Digital

Digital (61, 102, 108)-net over F₃, using

trace code for nets [i] based on digital (10, 51, 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]

(61, 61+41, 1046)-Net in Base 3 — Upper bound on s

There is no (61, 102, 1047)-net in base 3, because

1 times m-reduction [i] would yield (61, 101, 1047)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 557719 042518 266783 630167 702302 521176 189084 517721 > 3¹⁰¹ [i]