MinT - Best Known (120−51, 120, s)-Nets in Base 9

Best Known (120−51, 120, s)-Nets in Base 9

(120−51, 120, 344)-Net over F₉ — Constructive and digital

Digital (69, 120, 344)-net over F₉, using

4 times m-reduction [i] based on digital (69, 124, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 62, 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]

(120−51, 120, 488)-Net over F₉ — Digital

Digital (69, 120, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 60, 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]

(120−51, 120, 44318)-Net in Base 9 — Upper bound on s

There is no (69, 120, 44319)-net in base 9, because

1 times m-reduction [i] would yield (69, 119, 44319)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 358969 173629 423224 231662 074172 433341 887984 092228 763894 899527 023737 073557 322321 119840 695514 345535 014842 704491 941945 > 9¹¹⁹ [i]