MinT - Best Known (29, 29+109, s)-Nets in Base 9

Best Known (29, 29+109, s)-Nets in Base 9

(29, 29+109, 78)-Net over F₉ — Constructive and digital

Digital (29, 138, 78)-net over F₉, using

t-expansion [i] based on digital (22, 138, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(29, 29+109, 110)-Net over F₉ — Digital

Digital (29, 138, 110)-net over F₉, using

t-expansion [i] based on digital (26, 138, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(29, 29+109, 658)-Net in Base 9 — Upper bound on s

There is no (29, 138, 659)-net in base 9, because

1 times m-reduction [i] would yield (29, 137, 659)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 57840 176413 742447 961799 101909 601516 103952 479965 685848 049022 843968 641428 109402 155691 973084 819062 404185 940894 750973 196719 184603 663505 > 9¹³⁷ [i]