MinT - Best Known (66−37, 66, s)-Nets in Base 9

Best Known (66−37, 66, s)-Nets in Base 9

(66−37, 66, 78)-Net over F₉ — Constructive and digital

Digital (29, 66, 78)-net over F₉, using

t-expansion [i] based on digital (22, 66, 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]

(66−37, 66, 82)-Net in Base 9 — Constructive

(29, 66, 82)-net in base 9, using

base change [i] based on digital (7, 44, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(66−37, 66, 110)-Net over F₉ — Digital

Digital (29, 66, 110)-net over F₉, using

t-expansion [i] based on digital (26, 66, 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]

(66−37, 66, 2625)-Net in Base 9 — Upper bound on s

There is no (29, 66, 2626)-net in base 9, because

1 times m-reduction [i] would yield (29, 65, 2626)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 106 777725 597099 662814 409622 272920 021373 306712 459314 843255 447521 > 9⁶⁵ [i]