MinT - Best Known (139−85, 139, s)-Nets in Base 9

Best Known (139−85, 139, s)-Nets in Base 9

(139−85, 139, 81)-Net over F₉ — Constructive and digital

Digital (54, 139, 81)-net over F₉, using

t-expansion [i] based on digital (32, 139, 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]

(139−85, 139, 82)-Net in Base 9 — Constructive

(54, 139, 82)-net in base 9, using

2 times m-reduction [i] based on (54, 141, 82)-net in base 9, using
- base change [i] based on digital (7, 94, 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]

(139−85, 139, 182)-Net over F₉ — Digital

Digital (54, 139, 182)-net over F₉, using

t-expansion [i] based on digital (50, 139, 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]

(139−85, 139, 2793)-Net in Base 9 — Upper bound on s

There is no (54, 139, 2794)-net in base 9, because

1 times m-reduction [i] would yield (54, 138, 2794)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 490345 267866 800379 487998 551331 659977 208600 578275 966933 616087 564575 314496 248358 453717 816321 842471 881863 695672 961050 445419 881737 227745 > 9¹³⁸ [i]