MinT - Best Known (68, 68+105, s)-Nets in Base 8

Best Known (68, 68+105, s)-Nets in Base 8

(68, 68+105, 98)-Net over F₈ — Constructive and digital

Digital (68, 173, 98)-net over F₈, using

t-expansion [i] based on digital (37, 173, 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+105, 144)-Net over F₈ — Digital

Digital (68, 173, 144)-net over F₈, using

t-expansion [i] based on digital (45, 173, 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+105, 2772)-Net in Base 8 — Upper bound on s

There is no (68, 173, 2773)-net in base 8, because

1 times m-reduction [i] would yield (68, 172, 2773)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 217517 442954 082426 699172 669958 651215 212125 961247 666401 233719 315071 555005 325769 198931 961715 167964 522389 299389 229948 076844 720258 210539 250746 163609 030778 767158 > 8¹⁷² [i]