MinT - Best Known (126−49, 126, s)-Nets in Base 8

Best Known (126−49, 126, s)-Nets in Base 8

(126−49, 126, 354)-Net over F₈ — Constructive and digital

Digital (77, 126, 354)-net over F₈, using

14 times m-reduction [i] based on digital (77, 140, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 70, 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]

(126−49, 126, 432)-Net in Base 8 — Constructive

(77, 126, 432)-net in base 8, using

trace code for nets [i] based on (14, 63, 216)-net in base 64, using
- base change [i] based on digital (5, 54, 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]

(126−49, 126, 637)-Net over F₈ — Digital

Digital (77, 126, 637)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−49, 126, 70756)-Net in Base 8 — Upper bound on s

There is no (77, 126, 70757)-net in base 8, because

1 times m-reduction [i] would yield (77, 125, 70757)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 76968 297254 670285 057470 516115 229325 566969 253825 564683 464553 019067 420459 219805 167781 675753 897134 514241 191905 066524 > 8¹²⁵ [i]