MinT - Best Known (37, 37+51, s)-Nets in Base 9

Best Known (37, 37+51, s)-Nets in Base 9

(37, 37+51, 81)-Net over F₉ — Constructive and digital

Digital (37, 88, 81)-net over F₉, using

t-expansion [i] based on digital (32, 88, 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]

(37, 37+51, 82)-Net in Base 9 — Constructive

(37, 88, 82)-net in base 9, using

2 times m-reduction [i] based on (37, 90, 82)-net in base 9, using
- base change [i] based on digital (7, 60, 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]

(37, 37+51, 128)-Net over F₉ — Digital

Digital (37, 88, 128)-net over F₉, using

t-expansion [i] based on digital (33, 88, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(37, 37+51, 2647)-Net in Base 9 — Upper bound on s

There is no (37, 88, 2648)-net in base 9, because

1 times m-reduction [i] would yield (37, 87, 2648)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 105024 017512 119073 931437 619744 967801 697639 856221 601440 878849 549169 403956 057823 369409 > 9⁸⁷ [i]