MinT - Best Known (108−61, 108, s)-Nets in Base 9

Best Known (108−61, 108, s)-Nets in Base 9

(108−61, 108, 94)-Net over F₉ — Constructive and digital

Digital (47, 108, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 34, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (13, 74, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(108−61, 108, 96)-Net in Base 9 — Constructive

(47, 108, 96)-net in base 9, using

base change [i] based on digital (11, 72, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(108−61, 108, 162)-Net over F₉ — Digital

Digital (47, 108, 162)-net over F₉, using

t-expansion [i] based on digital (46, 108, 162)-net over F₉, using
- net from sequence [i] based on digital (46, 161)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
    - a curve from the manYPoints database [i]

(108−61, 108, 3793)-Net in Base 9 — Upper bound on s

There is no (47, 108, 3794)-net in base 9, because

1 times m-reduction [i] would yield (47, 107, 3794)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 271722 846433 690675 776284 986231 894118 882180 318593 154717 723257 709486 568639 857928 157935 489405 819978 965281 > 9¹⁰⁷ [i]