MinT - Best Known (42−19, 42, s)-Nets in Base 9

Best Known (42−19, 42, s)-Nets in Base 9

(42−19, 42, 232)-Net over F₉ — Constructive and digital

Digital (23, 42, 232)-net over F₉, using

trace code for nets [i] based on digital (2, 21, 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]

(42−19, 42, 236)-Net over F₉ — Digital

Digital (23, 42, 236)-net over F₉, using

trace code for nets [i] based on digital (2, 21, 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]

(42−19, 42, 11523)-Net in Base 9 — Upper bound on s

There is no (23, 42, 11524)-net in base 9, because

1 times m-reduction [i] would yield (23, 41, 11524)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1331 109350 198806 139685 846707 596333 171489 > 9⁴¹ [i]