MinT - Best Known (84, 84+75, s)-Nets in Base 8

Best Known (84, 84+75, s)-Nets in Base 8

(84, 84+75, 208)-Net over F₈ — Constructive and digital

Digital (84, 159, 208)-net over F₈, using

3 times m-reduction [i] based on digital (84, 162, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 81, 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]

(84, 84+75, 225)-Net in Base 8 — Constructive

(84, 159, 225)-net in base 8, using

t-expansion [i] based on (83, 159, 225)-net in base 8, using
- 13 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]

(84, 84+75, 326)-Net over F₈ — Digital

Digital (84, 159, 326)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 84+75, 15016)-Net in Base 8 — Upper bound on s

There is no (84, 159, 15017)-net in base 8, because

1 times m-reduction [i] would yield (84, 158, 15017)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48792 608974 207313 702677 452799 723375 415500 399064 803248 453649 332768 994867 720809 710198 997341 932872 365278 949278 053180 861630 707041 277932 681372 669176 > 8¹⁵⁸ [i]