MinT - Best Known (65, 65+47, s)-Nets in Base 9

Best Known (65, 65+47, s)-Nets in Base 9

(65, 65+47, 344)-Net over F₉ — Constructive and digital

Digital (65, 112, 344)-net over F₉, using

4 times m-reduction [i] based on digital (65, 116, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 58, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(65, 65+47, 488)-Net over F₉ — Digital

Digital (65, 112, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 56, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(65, 65+47, 47478)-Net in Base 9 — Upper bound on s

There is no (65, 112, 47479)-net in base 9, because

1 times m-reduction [i] would yield (65, 111, 47479)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8337 380273 917573 075704 815357 911094 599538 921033 483030 081050 637298 862926 346721 301711 044928 669284 951442 926217 > 9¹¹¹ [i]