MinT - Best Known (89−52, 89, s)-Nets in Base 9

Best Known (89−52, 89, s)-Nets in Base 9

(89−52, 89, 81)-Net over F₉ — Constructive and digital

Digital (37, 89, 81)-net over F₉, using

t-expansion [i] based on digital (32, 89, 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]

(89−52, 89, 82)-Net in Base 9 — Constructive

(37, 89, 82)-net in base 9, using

1 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]

(89−52, 89, 128)-Net over F₉ — Digital

Digital (37, 89, 128)-net over F₉, using

t-expansion [i] based on digital (33, 89, 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]

(89−52, 89, 2420)-Net in Base 9 — Upper bound on s

There is no (37, 89, 2421)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 544092 339538 057079 396828 587222 092173 598042 189444 514508 609962 192393 165103 162497 213393 > 9⁸⁹ [i]