MinT - Best Known (102−59, 102, s)-Nets in Base 9

Best Known (102−59, 102, s)-Nets in Base 9

(102−59, 102, 81)-Net over F₉ — Constructive and digital

Digital (43, 102, 81)-net over F₉, using

t-expansion [i] based on digital (32, 102, 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]

(102−59, 102, 88)-Net in Base 9 — Constructive

(43, 102, 88)-net in base 9, using

base change [i] based on digital (9, 68, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(102−59, 102, 147)-Net over F₉ — Digital

Digital (43, 102, 147)-net over F₉, using

net from sequence [i] based on digital (43, 146)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
  - a curve from the manYPoints database [i]

(102−59, 102, 3054)-Net in Base 9 — Upper bound on s

There is no (43, 102, 3055)-net in base 9, because

1 times m-reduction [i] would yield (43, 101, 3055)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 402843 345114 810923 011228 928331 266381 973398 192079 843285 364267 461124 474708 645270 750747 685831 253145 > 9¹⁰¹ [i]