MinT - Best Known (51−15, 51, s)-Nets in Base 9

Best Known (51−15, 51, s)-Nets in Base 9

(51−15, 51, 432)-Net over F₉ — Constructive and digital

Digital (36, 51, 432)-net over F₉, using

9¹ times duplication [i] based on digital (35, 50, 432)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 16, 200)-net over F₉, using
    - trace code for nets [i] based on digital (1, 8, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
  2. digital (19, 34, 232)-net over F₉, using
    - trace code for nets [i] based on digital (2, 17, 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]

(51−15, 51, 2269)-Net over F₉ — Digital

Digital (36, 51, 2269)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−15, 51, 2765960)-Net in Base 9 — Upper bound on s

There is no (36, 51, 2765961)-net in base 9, because

1 times m-reduction [i] would yield (36, 50, 2765961)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 515378 144266 901830 013869 588242 280965 288762 001465 > 9⁵⁰ [i]