MinT - Best Known (167−63, 167, s)-Nets in Base 8

Best Known (167−63, 167, s)-Nets in Base 8

(167−63, 167, 354)-Net over F₈ — Constructive and digital

Digital (104, 167, 354)-net over F₈, using

t-expansion [i] based on digital (93, 167, 354)-net over F₈, using
- 5 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]

(167−63, 167, 432)-Net in Base 8 — Constructive

(104, 167, 432)-net in base 8, using

5 times m-reduction [i] based on (104, 172, 432)-net in base 8, using
- trace code for nets [i] based on (18, 86, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
    - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(167−63, 167, 932)-Net over F₈ — Digital

Digital (104, 167, 932)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−63, 167, 121553)-Net in Base 8 — Upper bound on s

There is no (104, 167, 121554)-net in base 8, because

1 times m-reduction [i] would yield (104, 166, 121554)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 818494 888410 763245 521783 527028 942474 483825 841262 029404 280164 045590 267768 788960 472393 882419 173941 768466 059959 013927 687320 535987 903868 527164 570575 170464 > 8¹⁶⁶ [i]