MinT - Best Known (51, 51+31, s)-Nets in Base 3

Best Known (51, 51+31, s)-Nets in Base 3

(51, 51+31, 80)-Net over F₃ — Constructive and digital

Digital (51, 82, 80)-net over F₃, using

4 times m-reduction [i] based on digital (51, 86, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 43, 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]

(51, 51+31, 108)-Net over F₃ — Digital

Digital (51, 82, 108)-net over F₃, using

trace code for nets [i] based on digital (10, 41, 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]

(51, 51+31, 1196)-Net in Base 3 — Upper bound on s

There is no (51, 82, 1197)-net in base 3, because

1 times m-reduction [i] would yield (51, 81, 1197)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 445 054389 620987 674134 211514 223070 113051 > 3⁸¹ [i]