MinT - Best Known (166−65, 166, s)-Nets in Base 8

Best Known (166−65, 166, s)-Nets in Base 8

(166−65, 166, 354)-Net over F₈ — Constructive and digital

Digital (101, 166, 354)-net over F₈, using

t-expansion [i] based on digital (93, 166, 354)-net over F₈, using
- 6 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]

(166−65, 166, 432)-Net in Base 8 — Constructive

(101, 166, 432)-net in base 8, using

2 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]

(166−65, 166, 771)-Net over F₈ — Digital

Digital (101, 166, 771)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−65, 166, 82839)-Net in Base 8 — Upper bound on s

There is no (101, 166, 82840)-net in base 8, because

1 times m-reduction [i] would yield (101, 165, 82840)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102332 831289 611214 853893 956771 067165 991979 741128 049580 994484 751087 684883 300931 716138 717337 333777 650723 223549 497289 065225 846928 703537 278641 188337 855308 > 8¹⁶⁵ [i]