MinT - Best Known (86, 86+83, s)-Nets in Base 8

Best Known (86, 86+83, s)-Nets in Base 8

(86, 86+83, 194)-Net over F₈ — Constructive and digital

Digital (86, 169, 194)-net over F₈, using

t-expansion [i] based on digital (85, 169, 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+83, 225)-Net in Base 8 — Constructive

(86, 169, 225)-net in base 8, using

t-expansion [i] based on (83, 169, 225)-net in base 8, using
- 3 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+83, 291)-Net over F₈ — Digital

Digital (86, 169, 291)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+83, 11542)-Net in Base 8 — Upper bound on s

There is no (86, 169, 11543)-net in base 8, because

1 times m-reduction [i] would yield (86, 168, 11543)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 417029 496284 528022 953225 719359 477938 734077 957218 904942 378000 620897 191508 677561 501139 802366 879352 988306 754459 001035 960906 611874 675698 238430 424334 585440 > 8¹⁶⁸ [i]