MinT - Best Known (166−84, 166, s)-Nets in Base 8

Best Known (166−84, 166, s)-Nets in Base 8

(166−84, 166, 130)-Net over F₈ — Constructive and digital

Digital (82, 166, 130)-net over F₈, using

t-expansion [i] based on digital (76, 166, 130)-net over F₈, using
- 6 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(166−84, 166, 225)-Net in Base 8 — Constructive

(82, 166, 225)-net in base 8, using

2 times m-reduction [i] based on (82, 168, 225)-net in base 8, using
- base change [i] based on digital (40, 126, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(166−84, 166, 254)-Net over F₈ — Digital

Digital (82, 166, 254)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−84, 166, 8724)-Net in Base 8 — Upper bound on s

There is no (82, 166, 8725)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 820074 742822 319921 131661 476098 146794 771072 415065 556763 613334 845210 403117 758208 420249 421584 144493 092262 253007 618894 336933 954440 768956 600728 649262 231744 > 8¹⁶⁶ [i]