MinT - Best Known (131−79, 131, s)-Nets in Base 9

Best Known (131−79, 131, s)-Nets in Base 9

(131−79, 131, 81)-Net over F₉ — Constructive and digital

Digital (52, 131, 81)-net over F₉, using

t-expansion [i] based on digital (32, 131, 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]

(131−79, 131, 84)-Net in Base 9 — Constructive

(52, 131, 84)-net in base 9, using

1 times m-reduction [i] based on (52, 132, 84)-net in base 9, using
- base change [i] based on digital (8, 88, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(131−79, 131, 182)-Net over F₉ — Digital

Digital (52, 131, 182)-net over F₉, using

t-expansion [i] based on digital (50, 131, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(131−79, 131, 2894)-Net in Base 9 — Upper bound on s

There is no (52, 131, 2895)-net in base 9, because

1 times m-reduction [i] would yield (52, 130, 2895)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11323 920345 269212 329898 712629 073455 384305 029032 832751 052736 412471 183452 971473 867010 364131 152663 203807 473395 224027 493037 901385 > 9¹³⁰ [i]