MinT - Best Known (126−65, 126, s)-Nets in Base 9

Best Known (126−65, 126, s)-Nets in Base 9

(126−65, 126, 128)-Net over F₉ — Constructive and digital

Digital (61, 126, 128)-net over F₉, using

5 times m-reduction [i] based on digital (61, 131, 128)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 48, 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, 83, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(126−65, 126, 209)-Net over F₉ — Digital

Digital (61, 126, 209)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−65, 126, 8517)-Net in Base 9 — Upper bound on s

There is no (61, 126, 8518)-net in base 9, because

1 times m-reduction [i] would yield (61, 125, 8518)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 191103 818119 916408 116549 768430 147566 160276 460752 297595 911790 213115 876429 203999 838196 347801 952244 498946 474910 189119 992321 > 9¹²⁵ [i]