MinT - Best Known (51, 51+68, s)-Nets in Base 9

Best Known (51, 51+68, s)-Nets in Base 9

(51, 51+68, 94)-Net over F₉ — Constructive and digital

Digital (51, 119, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 38, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (13, 81, 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]

(51, 51+68, 96)-Net in Base 9 — Constructive

(51, 119, 96)-net in base 9, using

1 times m-reduction [i] based on (51, 120, 96)-net in base 9, using
- base change [i] based on digital (11, 80, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(51, 51+68, 182)-Net over F₉ — Digital

Digital (51, 119, 182)-net over F₉, using

t-expansion [i] based on digital (50, 119, 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]

(51, 51+68, 3679)-Net in Base 9 — Upper bound on s

There is no (51, 119, 3680)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 360254 769313 913274 016447 092820 558046 406201 739305 735170 464441 527753 204436 017714 870146 872256 764961 264460 065784 451585 > 9¹¹⁹ [i]