MinT - Best Known (85, 85+77, s)-Nets in Base 8

Best Known (85, 85+77, s)-Nets in Base 8

(85, 85+77, 208)-Net over F₈ — Constructive and digital

Digital (85, 162, 208)-net over F₈, using

2 times m-reduction [i] based on digital (85, 164, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 82, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(85, 85+77, 225)-Net in Base 8 — Constructive

(85, 162, 225)-net in base 8, using

t-expansion [i] based on (83, 162, 225)-net in base 8, using
- 10 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+77, 321)-Net over F₈ — Digital

Digital (85, 162, 321)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+77, 14362)-Net in Base 8 — Upper bound on s

There is no (85, 162, 14363)-net in base 8, because

1 times m-reduction [i] would yield (85, 161, 14363)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 002212 217605 995372 133249 193122 675625 206265 615980 963927 850656 599180 023671 862802 016151 100494 753773 124653 302945 869251 957770 357982 428564 066036 680832 > 8¹⁶¹ [i]