MinT - Best Known (82, 82+49, s)-Nets in Base 8

Best Known (82, 82+49, s)-Nets in Base 8

(82, 82+49, 354)-Net over F₈ — Constructive and digital

Digital (82, 131, 354)-net over F₈, using

19 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+49, 432)-Net in Base 8 — Constructive

(82, 131, 432)-net in base 8, using

3 times m-reduction [i] based on (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+49, 800)-Net over F₈ — Digital

Digital (82, 131, 800)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+49, 109129)-Net in Base 8 — Upper bound on s

There is no (82, 131, 109130)-net in base 8, because

1 times m-reduction [i] would yield (82, 130, 109130)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2521 928057 509218 192578 184687 153437 450978 503962 881173 677156 487253 370463 568511 557927 108890 856688 550918 513746 477794 543876 > 8¹³⁰ [i]