MinT - Best Known (128−65, 128, s)-Nets in Base 9

Best Known (128−65, 128, s)-Nets in Base 9

(128−65, 128, 138)-Net over F₉ — Constructive and digital

Digital (63, 128, 138)-net over F₉, using

1 times m-reduction [i] based on digital (63, 129, 138)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 46, 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 (17, 83, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(128−65, 128, 226)-Net over F₉ — Digital

Digital (63, 128, 226)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−65, 128, 9774)-Net in Base 9 — Upper bound on s

There is no (63, 128, 9775)-net in base 9, because

1 times m-reduction [i] would yield (63, 127, 9775)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 492597 157354 366896 471770 532158 893855 117287 903642 213791 144311 486349 417400 045433 428701 050289 964756 913372 736975 799818 686209 > 9¹²⁷ [i]