MinT - Best Known (135−91, 135, s)-Nets in Base 9

Best Known (135−91, 135, s)-Nets in Base 9

(135−91, 135, 81)-Net over F₉ — Constructive and digital

Digital (44, 135, 81)-net over F₉, using

t-expansion [i] based on digital (32, 135, 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]

(135−91, 135, 147)-Net over F₉ — Digital

Digital (44, 135, 147)-net over F₉, using

t-expansion [i] based on digital (43, 135, 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]

(135−91, 135, 1502)-Net in Base 9 — Upper bound on s

There is no (44, 135, 1503)-net in base 9, because

1 times m-reduction [i] would yield (44, 134, 1503)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 75 593065 585627 241144 318626 933180 194143 659474 780080 774339 157172 759367 667085 216636 770121 107315 358273 382858 036729 052846 979471 636889 > 9¹³⁴ [i]