MinT - Best Known (75, 75+57, s)-Nets in Base 9

Best Known (75, 75+57, s)-Nets in Base 9

(75, 75+57, 344)-Net over F₉ — Constructive and digital

Digital (75, 132, 344)-net over F₉, using

4 times m-reduction [i] based on digital (75, 136, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 68, 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]

(75, 75+57, 488)-Net over F₉ — Digital

Digital (75, 132, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 66, 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]

(75, 75+57, 41136)-Net in Base 9 — Upper bound on s

There is no (75, 132, 41137)-net in base 9, because

1 times m-reduction [i] would yield (75, 131, 41137)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 101371 456963 583140 132388 091916 864612 835485 963935 814529 429426 298026 902381 537857 477749 434090 134033 390720 933474 200647 627078 104417 > 9¹³¹ [i]