MinT - Best Known (167−65, 167, s)-Nets in Base 8

Best Known (167−65, 167, s)-Nets in Base 8

(167−65, 167, 354)-Net over F₈ — Constructive and digital

Digital (102, 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]

(167−65, 167, 432)-Net in Base 8 — Constructive

(102, 167, 432)-net in base 8, using

t-expansion [i] based on (101, 167, 432)-net in base 8, using
- 1 times m-reduction [i] based on (101, 168, 432)-net in base 8, using
  - trace code for nets [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(167−65, 167, 798)-Net over F₈ — Digital

Digital (102, 167, 798)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−65, 167, 88402)-Net in Base 8 — Upper bound on s

There is no (102, 167, 88403)-net in base 8, because

1 times m-reduction [i] would yield (102, 166, 88403)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 818586 156243 496672 739433 868625 357239 324911 093522 725626 084857 780699 781634 964660 899518 663212 506945 804737 728859 075503 932489 324804 028656 856009 505507 857690 > 8¹⁶⁶ [i]