MinT - Best Known (85, 85+81, s)-Nets in Base 8

Best Known (85, 85+81, s)-Nets in Base 8

(85, 85+81, 194)-Net over F₈ — Constructive and digital

Digital (85, 166, 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]

(85, 85+81, 225)-Net in Base 8 — Constructive

(85, 166, 225)-net in base 8, using

t-expansion [i] based on (83, 166, 225)-net in base 8, using
- 6 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]

(85, 85+81, 294)-Net over F₈ — Digital

Digital (85, 166, 294)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+81, 11940)-Net in Base 8 — Upper bound on s

There is no (85, 166, 11941)-net in base 8, because

1 times m-reduction [i] would yield (85, 165, 11941)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102452 851421 718801 418129 575471 429328 728917 962706 532372 665054 617586 907833 448100 233092 083009 561864 089226 671063 933182 177121 047373 355145 104063 916065 222800 > 8¹⁶⁵ [i]