MinT - Best Known (162−63, 162, s)-Nets in Base 8

Best Known (162−63, 162, s)-Nets in Base 8

(162−63, 162, 354)-Net over F₈ — Constructive and digital

Digital (99, 162, 354)-net over F₈, using

t-expansion [i] based on digital (93, 162, 354)-net over F₈, using
- 10 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]

(162−63, 162, 432)-Net in Base 8 — Constructive

(99, 162, 432)-net in base 8, using

2 times m-reduction [i] based on (99, 164, 432)-net in base 8, using
- trace code for nets [i] based on (17, 82, 216)-net in base 64, using
  - 2 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(162−63, 162, 781)-Net over F₈ — Digital

Digital (99, 162, 781)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−63, 162, 86911)-Net in Base 8 — Upper bound on s

There is no (99, 162, 86912)-net in base 8, because

1 times m-reduction [i] would yield (99, 161, 86912)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 24 974602 187139 812932 599865 832046 998653 021796 240456 213728 232274 010886 403236 998857 367994 743090 437826 761375 898403 944877 378756 563417 336253 383793 126025 > 8¹⁶¹ [i]