MinT - Best Known (57, 57+89, s)-Nets in Base 8

Best Known (57, 57+89, s)-Nets in Base 8

(57, 57+89, 98)-Net over F₈ — Constructive and digital

Digital (57, 146, 98)-net over F₈, using

t-expansion [i] based on digital (37, 146, 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]

(57, 57+89, 144)-Net over F₈ — Digital

Digital (57, 146, 144)-net over F₈, using

t-expansion [i] based on digital (45, 146, 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]

(57, 57+89, 2305)-Net in Base 8 — Upper bound on s

There is no (57, 146, 2306)-net in base 8, because

1 times m-reduction [i] would yield (57, 145, 2306)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 89862 137124 957726 479319 948143 086642 724820 885767 724845 507698 651429 483195 285635 209124 288432 156253 416809 372818 957727 517496 973667 849792 > 8¹⁴⁵ [i]