MinT - Best Known (101−57, 101, s)-Nets in Base 9

Best Known (101−57, 101, s)-Nets in Base 9

(101−57, 101, 92)-Net over F₉ — Constructive and digital

Digital (44, 101, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (13, 70, 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]

(101−57, 101, 94)-Net in Base 9 — Constructive

(44, 101, 94)-net in base 9, using

1 times m-reduction [i] based on (44, 102, 94)-net in base 9, using
- base change [i] based on digital (10, 68, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(101−57, 101, 147)-Net over F₉ — Digital

Digital (44, 101, 147)-net over F₉, using

t-expansion [i] based on digital (43, 101, 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]

(101−57, 101, 3596)-Net in Base 9 — Upper bound on s

There is no (44, 101, 3597)-net in base 9, because

1 times m-reduction [i] would yield (44, 100, 3597)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 266665 753679 575835 773695 008683 349703 073904 727554 599291 867600 532858 337391 194491 398634 694819 367649 > 9¹⁰⁰ [i]