MinT - Best Known (99, 99+69, s)-Nets in Base 8

Best Known (99, 99+69, s)-Nets in Base 8

(99, 99+69, 354)-Net over F₈ — Constructive and digital

Digital (99, 168, 354)-net over F₈, using

t-expansion [i] based on digital (93, 168, 354)-net over F₈, using
- 4 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(99, 99+69, 384)-Net in Base 8 — Constructive

(99, 168, 384)-net in base 8, using

trace code for nets [i] based on (15, 84, 192)-net in base 64, using
- base change [i] based on digital (3, 72, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(99, 99+69, 622)-Net over F₈ — Digital

Digital (99, 168, 622)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+69, 52718)-Net in Base 8 — Upper bound on s

There is no (99, 168, 52719)-net in base 8, because

1 times m-reduction [i] would yield (99, 167, 52719)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 546848 213816 801781 137691 549808 383760 358122 594605 855506 401960 878972 560508 743290 274872 698587 109655 675586 358898 354721 460139 298373 493472 863397 651519 216453 > 8¹⁶⁷ [i]