MinT - Best Known (137−51, 137, s)-Nets in Base 8

Best Known (137−51, 137, s)-Nets in Base 8

(137−51, 137, 354)-Net over F₈ — Constructive and digital

Digital (86, 137, 354)-net over F₈, using

21 times m-reduction [i] based on digital (86, 158, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 79, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(137−51, 137, 514)-Net in Base 8 — Constructive

(86, 137, 514)-net in base 8, using

8¹ times duplication [i] based on (85, 136, 514)-net in base 8, using
- base change [i] based on digital (51, 102, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 51, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(137−51, 137, 849)-Net over F₈ — Digital

Digital (86, 137, 849)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−51, 137, 118926)-Net in Base 8 — Upper bound on s

There is no (86, 137, 118927)-net in base 8, because

1 times m-reduction [i] would yield (86, 136, 118927)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 661 056145 112347 054711 633082 860797 303587 496350 213074 615269 277872 568439 043841 271819 955611 328962 497293 013466 312122 270580 874992 > 8¹³⁶ [i]