MinT - Best Known (63−32, 63, s)-Nets in Base 9

Best Known (63−32, 63, s)-Nets in Base 9

(63−32, 63, 84)-Net over F₉ — Constructive and digital

Digital (31, 63, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 45, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(63−32, 63, 94)-Net in Base 9 — Constructive

(31, 63, 94)-net in base 9, using

base change [i] based on digital (10, 42, 94)-net over F₂₇, using
- net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
    - a function field by SÃ©mirat [i]

(63−32, 63, 129)-Net over F₉ — Digital

Digital (31, 63, 129)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−32, 63, 4852)-Net in Base 9 — Upper bound on s

There is no (31, 63, 4853)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 313911 008575 689822 095802 675517 619833 944272 087994 391218 523265 > 9⁶³ [i]