MinT - Best Known (108−63, 108, s)-Nets in Base 9

Best Known (108−63, 108, s)-Nets in Base 9

(108−63, 108, 81)-Net over F₉ — Constructive and digital

Digital (45, 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]

(108−63, 108, 88)-Net in Base 9 — Constructive

(45, 108, 88)-net in base 9, using

base change [i] based on digital (9, 72, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(108−63, 108, 147)-Net over F₉ — Digital

Digital (45, 108, 147)-net over F₉, using

t-expansion [i] based on digital (43, 108, 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]

(108−63, 108, 3033)-Net in Base 9 — Upper bound on s

There is no (45, 108, 3034)-net in base 9, because

1 times m-reduction [i] would yield (45, 107, 3034)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 278190 700350 327562 827356 544200 908199 789255 201261 196506 838877 957399 490585 502387 989179 584754 285835 685617 > 9¹⁰⁷ [i]