MinT - Best Known (105−61, 105, s)-Nets in Base 9

Best Known (105−61, 105, s)-Nets in Base 9

(105−61, 105, 81)-Net over F₉ — Constructive and digital

Digital (44, 105, 81)-net over F₉, using

t-expansion [i] based on digital (32, 105, 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]

(105−61, 105, 88)-Net in Base 9 — Constructive

(44, 105, 88)-net in base 9, using

base change [i] based on digital (9, 70, 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]

(105−61, 105, 147)-Net over F₉ — Digital

Digital (44, 105, 147)-net over F₉, using

t-expansion [i] based on digital (43, 105, 147)-net over F₉, using
- net from sequence [i] based on digital (43, 146)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
    - a curve from the manYPoints database [i]

(105−61, 105, 3041)-Net in Base 9 — Upper bound on s

There is no (44, 105, 3042)-net in base 9, because

1 times m-reduction [i] would yield (44, 104, 3042)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1744 254445 927237 893186 609871 473763 177213 896401 527506 561424 714155 559234 218237 204507 312085 509490 400801 > 9¹⁰⁴ [i]