MinT - Best Known (59, 59+89, s)-Nets in Base 9

Best Known (59, 59+89, s)-Nets in Base 9

(59, 59+89, 84)-Net over F₉ — Constructive and digital

Digital (59, 148, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 46, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 102, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(59, 59+89, 88)-Net in Base 9 — Constructive

(59, 148, 88)-net in base 9, using

2 times m-reduction [i] based on (59, 150, 88)-net in base 9, using
- base change [i] based on digital (9, 100, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(59, 59+89, 182)-Net over F₉ — Digital

Digital (59, 148, 182)-net over F₉, using

t-expansion [i] based on digital (50, 148, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(59, 59+89, 3298)-Net in Base 9 — Upper bound on s

There is no (59, 148, 3299)-net in base 9, because

1 times m-reduction [i] would yield (59, 147, 3299)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 188 612718 900446 015950 651555 625851 990206 598183 699248 258544 163310 190206 724612 024816 848448 315111 109604 186853 251204 448512 339155 148823 311076 412193 > 9¹⁴⁷ [i]