MinT - Best Known (42, 42+31, s)-Nets in Base 9

Best Known (42, 42+31, s)-Nets in Base 9

(42, 42+31, 320)-Net over F₉ — Constructive and digital

Digital (42, 73, 320)-net over F₉, using

1 times m-reduction [i] based on digital (42, 74, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(42, 42+31, 334)-Net over F₉ — Digital

Digital (42, 73, 334)-net over F₉, using

1 times m-reduction [i] based on digital (42, 74, 334)-net over F₉, using
- trace code for nets [i] based on digital (5, 37, 167)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
      - a curve from the manYPoints database [i]

(42, 42+31, 30543)-Net in Base 9 — Upper bound on s

There is no (42, 73, 30544)-net in base 9, because

1 times m-reduction [i] would yield (42, 72, 30544)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 507 728187 000028 563282 182652 380999 781589 401377 696417 646796 003181 553537 > 9⁷² [i]