MinT - Best Known (89−55, 89, s)-Nets in Base 9

Best Known (89−55, 89, s)-Nets in Base 9

(89−55, 89, 81)-Net over F₉ — Constructive and digital

Digital (34, 89, 81)-net over F₉, using

t-expansion [i] based on digital (32, 89, 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]

(89−55, 89, 128)-Net over F₉ — Digital

Digital (34, 89, 128)-net over F₉, using

t-expansion [i] based on digital (33, 89, 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]

(89−55, 89, 1743)-Net in Base 9 — Upper bound on s

There is no (34, 89, 1744)-net in base 9, because

1 times m-reduction [i] would yield (34, 88, 1744)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 949481 673611 546593 475882 340382 122299 565998 410722 831942 762831 754066 369906 065547 617665 > 9⁸⁸ [i]