MinT - Best Known (85, 168, s)-Nets in Base 8

Best Known (85, 168, s)-Nets in Base 8

(85, 168, 194)-Net over F₈ — Constructive and digital

Digital (85, 168, 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, 168, 225)-Net in Base 8 — Constructive

(85, 168, 225)-net in base 8, using

t-expansion [i] based on (83, 168, 225)-net in base 8, using
- 4 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, 168, 282)-Net over F₈ — Digital

Digital (85, 168, 282)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 168, 10970)-Net in Base 8 — Upper bound on s

There is no (85, 168, 10971)-net in base 8, because

1 times m-reduction [i] would yield (85, 167, 10971)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 554999 171989 996414 276151 674488 290004 484099 502111 772494 366035 222130 836418 553041 783449 484761 118386 229851 972222 275903 484187 306458 727705 680738 888385 999292 > 8¹⁶⁷ [i]