MinT - Best Known (52, 125, s)-Nets in Base 9

Best Known (52, 125, s)-Nets in Base 9

(52, 125, 92)-Net over F₉ — Constructive and digital

Digital (52, 125, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 39, 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 (13, 86, 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]

(52, 125, 94)-Net in Base 9 — Constructive

(52, 125, 94)-net in base 9, using

1 times m-reduction [i] based on (52, 126, 94)-net in base 9, using
- base change [i] based on digital (10, 84, 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]

(52, 125, 182)-Net over F₉ — Digital

Digital (52, 125, 182)-net over F₉, using

t-expansion [i] based on digital (50, 125, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(52, 125, 3433)-Net in Base 9 — Upper bound on s

There is no (52, 125, 3434)-net in base 9, because

1 times m-reduction [i] would yield (52, 124, 3434)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21338 931460 839821 273152 921287 879276 242595 288543 181006 086515 216694 357298 518095 420378 514981 964139 138401 765922 210103 387585 > 9¹²⁴ [i]