MinT - Best Known (125−72, 125, s)-Nets in Base 9

Best Known (125−72, 125, s)-Nets in Base 9

(125−72, 125, 94)-Net over F₉ — Constructive and digital

Digital (53, 125, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 40, 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, 85, 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]

(125−72, 125, 96)-Net in Base 9 — Constructive

(53, 125, 96)-net in base 9, using

1 times m-reduction [i] based on (53, 126, 96)-net in base 9, using
- base change [i] based on digital (11, 84, 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]

(125−72, 125, 182)-Net over F₉ — Digital

Digital (53, 125, 182)-net over F₉, using

t-expansion [i] based on digital (50, 125, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(125−72, 125, 3650)-Net in Base 9 — Upper bound on s

There is no (53, 125, 3651)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 191114 215090 612708 112566 653619 120211 011624 285740 753606 544044 280189 571051 958215 504664 667626 883244 715579 385630 419549 572193 > 9¹²⁵ [i]