MinT - Best Known (68, 68+45, s)-Nets in Base 9

Best Known (68, 68+45, s)-Nets in Base 9

(68, 68+45, 344)-Net over F₉ — Constructive and digital

Digital (68, 113, 344)-net over F₉, using

9 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+45, 623)-Net over F₉ — Digital

Digital (68, 113, 623)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+45, 81593)-Net in Base 9 — Upper bound on s

There is no (68, 113, 81594)-net in base 9, because

1 times m-reduction [i] would yield (68, 112, 81594)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 75031 805572 966817 282432 976977 715849 307257 295678 397580 438746 826140 750086 787779 022352 130774 971222 285668 956705 > 9¹¹² [i]