MinT - Best Known (59−30, 59, s)-Nets in Base 9

Best Known (59−30, 59, s)-Nets in Base 9

(59−30, 59, 82)-Net over F₉ — Constructive and digital

Digital (29, 59, 82)-net over F₉, using

base reduction for projective spaces (embedding PG(29,81) in PG(58,9)) for nets [i] based on digital (0, 30, 82)-net over F₈₁, using
- net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
    - the rational function field F₈₁(x) [i]
  - Niederreiter sequence [i]

(59−30, 59, 88)-Net in Base 9 — Constructive

(29, 59, 88)-net in base 9, using

1 times m-reduction [i] based on (29, 60, 88)-net in base 9, using
- base change [i] based on digital (9, 40, 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]

(59−30, 59, 123)-Net over F₉ — Digital

Digital (29, 59, 123)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−30, 59, 4541)-Net in Base 9 — Upper bound on s

There is no (29, 59, 4542)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 200 108815 360337 258479 252458 674781 455956 457551 972621 222737 > 9⁵⁹ [i]