MinT - Best Known (138−83, 138, s)-Nets in Base 9

Best Known (138−83, 138, s)-Nets in Base 9

(138−83, 138, 81)-Net over F₉ — Constructive and digital

Digital (55, 138, 81)-net over F₉, using

t-expansion [i] based on digital (32, 138, 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]

(138−83, 138, 88)-Net in Base 9 — Constructive

(55, 138, 88)-net in base 9, using

base change [i] based on digital (9, 92, 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]

(138−83, 138, 182)-Net over F₉ — Digital

Digital (55, 138, 182)-net over F₉, using

t-expansion [i] based on digital (50, 138, 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]

(138−83, 138, 3089)-Net in Base 9 — Upper bound on s

There is no (55, 138, 3090)-net in base 9, because

1 times m-reduction [i] would yield (55, 137, 3090)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54252 055866 583888 029036 194876 360013 819588 153456 262178 284894 201229 931559 483163 844346 279281 925999 626169 673940 388137 219671 158789 662225 > 9¹³⁷ [i]