MinT - Best Known (51, 51+71, s)-Nets in Base 9

Best Known (51, 51+71, s)-Nets in Base 9

(51, 51+71, 92)-Net over F₉ — Constructive and digital

Digital (51, 122, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 38, 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, 84, 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+71, 94)-Net in Base 9 — Constructive

(51, 122, 94)-net in base 9, using

1 times m-reduction [i] based on (51, 123, 94)-net in base 9, using
- base change [i] based on digital (10, 82, 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]

(51, 51+71, 182)-Net over F₉ — Digital

Digital (51, 122, 182)-net over F₉, using

t-expansion [i] based on digital (50, 122, 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+71, 3439)-Net in Base 9 — Upper bound on s

There is no (51, 122, 3440)-net in base 9, because

1 times m-reduction [i] would yield (51, 121, 3440)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29 339269 634883 029374 535079 144211 408627 192785 264540 693715 718395 278919 586550 583978 868856 035090 613343 115173 382712 401025 > 9¹²¹ [i]