MinT - Best Known (109−55, 109, s)-Nets in Base 9

Best Known (109−55, 109, s)-Nets in Base 9

(109−55, 109, 128)-Net over F₉ — Constructive and digital

Digital (54, 109, 128)-net over F₉, using

1 times m-reduction [i] based on digital (54, 110, 128)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 41, 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]
  2. digital (13, 69, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(109−55, 109, 204)-Net over F₉ — Digital

Digital (54, 109, 204)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−55, 109, 8943)-Net in Base 9 — Upper bound on s

There is no (54, 109, 8944)-net in base 9, because

1 times m-reduction [i] would yield (54, 108, 8944)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 461520 625487 121342 594961 073611 240890 187783 471722 372178 815001 089157 877227 873826 758740 673263 540803 460225 > 9¹⁰⁸ [i]