MinT - Best Known (137−83, 137, s)-Nets in Base 9

Best Known (137−83, 137, s)-Nets in Base 9

(137−83, 137, 81)-Net over F₉ — Constructive and digital

Digital (54, 137, 81)-net over F₉, using

t-expansion [i] based on digital (32, 137, 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]

(137−83, 137, 84)-Net in Base 9 — Constructive

(54, 137, 84)-net in base 9, using

1 times m-reduction [i] based on (54, 138, 84)-net in base 9, using
- base change [i] based on digital (8, 92, 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]

(137−83, 137, 182)-Net over F₉ — Digital

Digital (54, 137, 182)-net over F₉, using

t-expansion [i] based on digital (50, 137, 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]

(137−83, 137, 2926)-Net in Base 9 — Upper bound on s

There is no (54, 137, 2927)-net in base 9, because

1 times m-reduction [i] would yield (54, 136, 2927)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5989 735485 810322 223505 118106 997713 720804 685578 451529 044692 135523 568359 533332 396582 022642 259586 440275 507189 394991 893130 273723 780665 > 9¹³⁶ [i]