MinT - Best Known (99, 164, s)-Nets in Base 8

Best Known (99, 164, s)-Nets in Base 8

(99, 164, 354)-Net over F₈ — Constructive and digital

Digital (99, 164, 354)-net over F₈, using

t-expansion [i] based on digital (93, 164, 354)-net over F₈, using
- 8 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, 164, 432)-Net in Base 8 — Constructive

(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]

(99, 164, 720)-Net over F₈ — Digital

Digital (99, 164, 720)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 164, 72740)-Net in Base 8 — Upper bound on s

There is no (99, 164, 72741)-net in base 8, because

1 times m-reduction [i] would yield (99, 163, 72741)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 585784 631286 671004 345112 721680 106551 506659 379252 824087 352661 357915 281282 727596 761507 912895 649458 422809 710686 035290 900149 506263 346359 069905 734841 > 8¹⁶³ [i]