MinT - Best Known (150−90, 150, s)-Nets in Base 9

Best Known (150−90, 150, s)-Nets in Base 9

(150−90, 150, 84)-Net over F₉ — Constructive and digital

Digital (60, 150, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 47, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 103, 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]

(150−90, 150, 94)-Net in Base 9 — Constructive

(60, 150, 94)-net in base 9, using

base change [i] based on digital (10, 100, 94)-net over F₂₇, using
- net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
    - a function field by SÃ©mirat [i]

(150−90, 150, 190)-Net over F₉ — Digital

Digital (60, 150, 190)-net over F₉, using

net from sequence [i] based on digital (60, 189)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 60 and N(F) ≥ 190, using
  - Drinfeld modules of rank 1 [i]

(150−90, 150, 3313)-Net in Base 9 — Upper bound on s

There is no (60, 150, 3314)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 137703 378731 269430 695063 443323 313130 166804 531236 756597 965874 842885 134618 354442 657345 899202 728768 827430 969326 397844 588281 801392 109426 216603 864785 > 9¹⁵⁰ [i]