MinT - Best Known (108−75, 108, s)-Nets in Base 9

Best Known (108−75, 108, s)-Nets in Base 9

(108−75, 108, 81)-Net over F₉ — Constructive and digital

Digital (33, 108, 81)-net over F₉, using

t-expansion [i] based on digital (32, 108, 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]

(108−75, 108, 128)-Net over F₉ — Digital

Digital (33, 108, 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]

(108−75, 108, 1030)-Net in Base 9 — Upper bound on s

There is no (33, 108, 1031)-net in base 9, because

1 times m-reduction [i] would yield (33, 107, 1031)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 296624 949733 601121 475267 128267 032862 272015 554737 453962 827640 023061 965914 796950 202314 639761 465938 316825 > 9¹⁰⁷ [i]