MinT - Best Known (45, 45+39, s)-Nets in Base 9

Best Known (45, 45+39, s)-Nets in Base 9

(45, 45+39, 232)-Net over F₉ — Constructive and digital

Digital (45, 84, 232)-net over F₉, using

2 times m-reduction [i] based on digital (45, 86, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 43, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(45, 45+39, 272)-Net over F₉ — Digital

Digital (45, 84, 272)-net over F₉, using

trace code for nets [i] based on digital (3, 42, 136)-net over F₈₁, using
- net from sequence [i] based on digital (3, 135)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 3 and N(F) ≥ 136, using
    - a curve from the manYPoints database [i]

(45, 45+39, 14598)-Net in Base 9 — Upper bound on s

There is no (45, 84, 14599)-net in base 9, because

1 times m-reduction [i] would yield (45, 83, 14599)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 928670 161125 475618 320148 005789 098265 086725 462516 801099 046497 877709 913778 000169 > 9⁸³ [i]