MinT - Best Known (75−25, 75, s)-Nets in Base 9

Best Known (75−25, 75, s)-Nets in Base 9

(75−25, 75, 348)-Net over F₉ — Constructive and digital

Digital (50, 75, 348)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (35, 60, 320)-net over F₉, using
  - trace code for nets [i] based on digital (5, 30, 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]

(75−25, 75, 1188)-Net over F₉ — Digital

Digital (50, 75, 1188)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−25, 75, 506711)-Net in Base 9 — Upper bound on s

There is no (50, 75, 506712)-net in base 9, because

1 times m-reduction [i] would yield (50, 74, 506712)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 41110 158101 696950 745590 536077 551781 377461 352407 468240 983925 353816 278785 > 9⁷⁴ [i]