MinT - Best Known (138−85, 138, s)-Nets in Base 9

Best Known (138−85, 138, s)-Nets in Base 9

(138−85, 138, 81)-Net over F₉ — Constructive and digital

Digital (53, 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]

(138−85, 138, 82)-Net in Base 9 — Constructive

(53, 138, 82)-net in base 9, using

base change [i] based on digital (7, 92, 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]

(138−85, 138, 182)-Net over F₉ — Digital

Digital (53, 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]

(138−85, 138, 2649)-Net in Base 9 — Upper bound on s

There is no (53, 138, 2650)-net in base 9, because

1 times m-reduction [i] would yield (53, 137, 2650)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54240 032576 403223 404577 186616 756180 185746 931383 563354 947321 237215 727163 484032 225178 618895 683168 199010 920918 607004 107323 164477 464289 > 9¹³⁷ [i]