MinT - Best Known (55, 55+87, s)-Nets in Base 9

Best Known (55, 55+87, s)-Nets in Base 9

(55, 55+87, 81)-Net over F₉ — Constructive and digital

Digital (55, 142, 81)-net over F₉, using

t-expansion [i] based on digital (32, 142, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(55, 55+87, 82)-Net in Base 9 — Constructive

(55, 142, 82)-net in base 9, using

2 times m-reduction [i] based on (55, 144, 82)-net in base 9, using
- base change [i] based on digital (7, 96, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(55, 55+87, 182)-Net over F₉ — Digital

Digital (55, 142, 182)-net over F₉, using

t-expansion [i] based on digital (50, 142, 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]

(55, 55+87, 2814)-Net in Base 9 — Upper bound on s

There is no (55, 142, 2815)-net in base 9, because

1 times m-reduction [i] would yield (55, 141, 2815)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 357 256821 558231 673655 380997 325282 440483 996614 017485 487005 598667 583029 135740 040405 749954 089880 836382 184734 093987 306043 002573 665020 683945 > 9¹⁴¹ [i]