MinT - Best Known (164−67, 164, s)-Nets in Base 8

Best Known (164−67, 164, s)-Nets in Base 8

(164−67, 164, 354)-Net over F₈ — Constructive and digital

Digital (97, 164, 354)-net over F₈, using

t-expansion [i] based on digital (93, 164, 354)-net over F₈, using
- 8 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(164−67, 164, 384)-Net in Base 8 — Constructive

(97, 164, 384)-net in base 8, using

trace code for nets [i] based on (15, 82, 192)-net in base 64, using
- 2 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
  - base change [i] based on digital (3, 72, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(164−67, 164, 625)-Net over F₈ — Digital

Digital (97, 164, 625)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−67, 164, 54300)-Net in Base 8 — Upper bound on s

There is no (97, 164, 54301)-net in base 8, because

1 times m-reduction [i] would yield (97, 163, 54301)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 355590 957380 178304 882324 857605 723133 203943 522624 164501 439646 797711 856134 564312 996858 247348 777179 834261 852264 499297 665249 451061 388816 121763 865472 > 8¹⁶³ [i]