MinT - Best Known (68, 68+50, s)-Nets in Base 9

Best Known (68, 68+50, s)-Nets in Base 9

(68, 68+50, 344)-Net over F₉ — Constructive and digital

Digital (68, 118, 344)-net over F₉, using

4 times m-reduction [i] based on digital (68, 122, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 61, 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]

(68, 68+50, 488)-Net over F₉ — Digital

Digital (68, 118, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 59, 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]

(68, 68+50, 40588)-Net in Base 9 — Upper bound on s

There is no (68, 118, 40589)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39889 394964 976153 636959 728414 386544 739862 860746 096040 982716 503038 942387 371444 321644 679464 419758 021781 205405 339433 > 9¹¹⁸ [i]