MinT - Best Known (55, 55+78, s)-Nets in Base 9

Best Known (55, 55+78, s)-Nets in Base 9

(55, 55+78, 92)-Net over F₉ — Constructive and digital

Digital (55, 133, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 42, 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, 91, 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]

(55, 55+78, 94)-Net in Base 9 — Constructive

(55, 133, 94)-net in base 9, using

2 times m-reduction [i] based on (55, 135, 94)-net in base 9, using
- base change [i] based on digital (10, 90, 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]

(55, 55+78, 182)-Net over F₉ — Digital

Digital (55, 133, 182)-net over F₉, using

t-expansion [i] based on digital (50, 133, 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]

(55, 55+78, 3431)-Net in Base 9 — Upper bound on s

There is no (55, 133, 3432)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 212028 809014 405355 050491 337970 676038 805334 435181 123101 852402 806817 737630 909610 967258 533968 819179 331409 980630 567990 510312 583105 > 9¹³³ [i]