MinT - Best Known (51, 51+81, s)-Nets in Base 9

Best Known (51, 51+81, s)-Nets in Base 9

(51, 51+81, 81)-Net over F₉ — Constructive and digital

Digital (51, 132, 81)-net over F₉, using

t-expansion [i] based on digital (32, 132, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(51, 51+81, 82)-Net in Base 9 — Constructive

(51, 132, 82)-net in base 9, using

base change [i] based on digital (7, 88, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(51, 51+81, 182)-Net over F₉ — Digital

Digital (51, 132, 182)-net over F₉, using

t-expansion [i] based on digital (50, 132, 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+81, 2604)-Net in Base 9 — Upper bound on s

There is no (51, 132, 2605)-net in base 9, because

1 times m-reduction [i] would yield (51, 131, 2605)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 101493 930073 799982 412312 706346 264324 518612 452168 407532 117167 613310 337831 787935 567978 635695 339621 401634 979420 073039 867415 554369 > 9¹³¹ [i]