MinT - Best Known (68, 68+99, s)-Nets in Base 8

Best Known (68, 68+99, s)-Nets in Base 8

(68, 68+99, 98)-Net over F₈ — Constructive and digital

Digital (68, 167, 98)-net over F₈, using

t-expansion [i] based on digital (37, 167, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(68, 68+99, 144)-Net over F₈ — Digital

Digital (68, 167, 144)-net over F₈, using

t-expansion [i] based on digital (45, 167, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(68, 68+99, 150)-Net in Base 8

(68, 167, 150)-net in base 8, using

1 times m-reduction [i] based on (68, 168, 150)-net in base 8, using
- base change [i] based on digital (26, 126, 150)-net over F₁₆, using
  - net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
      - a curve from the manYPoints database [i]

(68, 68+99, 3099)-Net in Base 8 — Upper bound on s

There is no (68, 167, 3100)-net in base 8, because

1 times m-reduction [i] would yield (68, 166, 3100)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 819479 643933 241138 274739 830796 996746 870530 196862 693564 603057 853007 694871 330609 647409 529107 866185 226930 920549 214204 551522 855653 983031 527691 622155 372704 > 8¹⁶⁶ [i]