MinT - Best Known (57, 57+61, s)-Nets in Base 9

Best Known (57, 57+61, s)-Nets in Base 9

(57, 57+61, 128)-Net over F₉ — Constructive and digital

Digital (57, 118, 128)-net over F₉, using

1 times m-reduction [i] based on digital (57, 119, 128)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 44, 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, 75, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(57, 57+61, 196)-Net over F₉ — Digital

Digital (57, 118, 196)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+61, 7911)-Net in Base 9 — Upper bound on s

There is no (57, 118, 7912)-net in base 9, because

1 times m-reduction [i] would yield (57, 117, 7912)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4442 479967 228208 622332 388685 948949 607683 762459 519020 411735 046330 527953 457254 785479 921017 929715 611585 560706 594945 > 9¹¹⁷ [i]