MinT - Best Known (66−31, 66, s)-Nets in Base 9

Best Known (66−31, 66, s)-Nets in Base 9

(66−31, 66, 232)-Net over F₉ — Constructive and digital

Digital (35, 66, 232)-net over F₉, using

trace code for nets [i] based on digital (2, 33, 116)-net over F₈₁, using
- net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
    - a function field by SÃ©mirat [i]

(66−31, 66, 236)-Net over F₉ — Digital

Digital (35, 66, 236)-net over F₉, using

trace code for nets [i] based on digital (2, 33, 118)-net over F₈₁, using
- net from sequence [i] based on digital (2, 117)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 118, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(66−31, 66, 10949)-Net in Base 9 — Upper bound on s

There is no (35, 66, 10950)-net in base 9, because

1 times m-reduction [i] would yield (35, 65, 10950)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 106 252335 783530 682269 412645 744613 460608 400002 055802 400813 702161 > 9⁶⁵ [i]