MinT - Best Known (128−55, 128, s)-Nets in Base 9

Best Known (128−55, 128, s)-Nets in Base 9

(128−55, 128, 344)-Net over F₉ — Constructive and digital

Digital (73, 128, 344)-net over F₉, using

4 times m-reduction [i] based on digital (73, 132, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 66, 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]

(128−55, 128, 488)-Net over F₉ — Digital

Digital (73, 128, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 64, 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]

(128−55, 128, 42036)-Net in Base 9 — Upper bound on s

There is no (73, 128, 42037)-net in base 9, because

1 times m-reduction [i] would yield (73, 127, 42037)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 455015 282880 221236 451596 679251 387732 663576 348278 317444 573485 454592 267875 291538 172465 629464 705962 712592 579867 361205 226745 > 9¹²⁷ [i]