MinT - Best Known (86, 86+81, s)-Nets in Base 8

Best Known (86, 86+81, s)-Nets in Base 8

(86, 86+81, 194)-Net over F₈ — Constructive and digital

Digital (86, 167, 194)-net over F₈, using

t-expansion [i] based on digital (85, 167, 194)-net over F₈, using
- net from sequence [i] based on digital (85, 193)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 85 and N(F) ≥ 194, using
    - F₆ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(86, 86+81, 225)-Net in Base 8 — Constructive

(86, 167, 225)-net in base 8, using

t-expansion [i] based on (83, 167, 225)-net in base 8, using
- 5 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 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]

(86, 86+81, 303)-Net over F₈ — Digital

Digital (86, 167, 303)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+81, 12579)-Net in Base 8 — Upper bound on s

There is no (86, 167, 12580)-net in base 8, because

1 times m-reduction [i] would yield (86, 166, 12580)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 820851 625200 545115 672529 481376 770572 172911 109923 709848 120312 457624 719046 700597 864107 547557 081136 848383 218596 128879 908114 534558 725693 333601 197770 475488 > 8¹⁶⁶ [i]