MinT - Best Known (137−79, 137, s)-Nets in Base 9

Best Known (137−79, 137, s)-Nets in Base 9

(137−79, 137, 98)-Net over F₉ — Constructive and digital

Digital (58, 137, 98)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 45, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (13, 92, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(137−79, 137, 182)-Net over F₉ — Digital

Digital (58, 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]

(137−79, 137, 4068)-Net in Base 9 — Upper bound on s

There is no (58, 137, 4069)-net in base 9, because

1 times m-reduction [i] would yield (58, 136, 4069)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6023 739416 541022 803465 502585 234507 927777 310858 547359 325401 296630 657618 042197 798070 493978 193505 763996 217106 004041 054190 607721 795545 > 9¹³⁶ [i]