MinT - Best Known (27, 27+83, s)-Nets in Base 9

Best Known (27, 27+83, s)-Nets in Base 9

(27, 27+83, 78)-Net over F₉ — Constructive and digital

Digital (27, 110, 78)-net over F₉, using

t-expansion [i] based on digital (22, 110, 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]

(27, 27+83, 110)-Net over F₉ — Digital

Digital (27, 110, 110)-net over F₉, using

t-expansion [i] based on digital (26, 110, 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]

(27, 27+83, 669)-Net in Base 9 — Upper bound on s

There is no (27, 110, 670)-net in base 9, because

1 times m-reduction [i] would yield (27, 109, 670)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 104 737790 022914 545493 833964 430715 936814 113945 264064 344320 194574 983633 300149 016689 278017 507869 977972 242289 > 9¹⁰⁹ [i]