MinT - Best Known (82, 82+52, s)-Nets in Base 8

Best Known (82, 82+52, s)-Nets in Base 8

(82, 82+52, 354)-Net over F₈ — Constructive and digital

Digital (82, 134, 354)-net over F₈, using

16 times m-reduction [i] based on digital (82, 150, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 75, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(82, 82+52, 432)-Net in Base 8 — Constructive

(82, 134, 432)-net in base 8, using

trace code for nets [i] based on (15, 67, 216)-net in base 64, using
- 3 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
  - base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(82, 82+52, 676)-Net over F₈ — Digital

Digital (82, 134, 676)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+52, 67995)-Net in Base 8 — Upper bound on s

There is no (82, 134, 67996)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 332123 575889 747721 192037 507233 225121 785094 306908 923673 587929 705448 697370 383924 879300 487733 190131 806717 721574 220924 030243 > 8¹³⁴ [i]