MinT - Best Known (138−101, 138, s)-Nets in Base 9

Best Known (138−101, 138, s)-Nets in Base 9

(138−101, 138, 81)-Net over F₉ — Constructive and digital

Digital (37, 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]

(138−101, 138, 128)-Net over F₉ — Digital

Digital (37, 138, 128)-net over F₉, using

t-expansion [i] based on digital (33, 138, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(138−101, 138, 972)-Net in Base 9 — Upper bound on s

There is no (37, 138, 973)-net in base 9, because

1 times m-reduction [i] would yield (37, 137, 973)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 55823 241161 734106 504340 457164 137028 874568 022956 531739 296815 627356 546970 081612 306452 915113 418408 831568 077474 490280 829228 085159 499025 > 9¹³⁷ [i]