MinT - Best Known (30, 30+109, s)-Nets in Base 9

Best Known (30, 30+109, s)-Nets in Base 9

(30, 30+109, 78)-Net over F₉ — Constructive and digital

Digital (30, 139, 78)-net over F₉, using

t-expansion [i] based on digital (22, 139, 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]

(30, 30+109, 110)-Net over F₉ — Digital

Digital (30, 139, 110)-net over F₉, using

t-expansion [i] based on digital (26, 139, 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]

(30, 30+109, 686)-Net in Base 9 — Upper bound on s

There is no (30, 139, 687)-net in base 9, because

1 times m-reduction [i] would yield (30, 138, 687)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 493513 915468 362639 942161 188516 430968 135435 574809 730261 308583 067674 879562 174248 679530 886841 463115 669873 079851 380975 589750 079843 773777 > 9¹³⁸ [i]