MinT - Best Known (171−111, 171, s)-Nets in Base 8

Best Known (171−111, 171, s)-Nets in Base 8

(171−111, 171, 98)-Net over F₈ — Constructive and digital

Digital (60, 171, 98)-net over F₈, using

t-expansion [i] based on digital (37, 171, 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]

(171−111, 171, 144)-Net over F₈ — Digital

Digital (60, 171, 144)-net over F₈, using

t-expansion [i] based on digital (45, 171, 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]

(171−111, 171, 1850)-Net in Base 8 — Upper bound on s

There is no (60, 171, 1851)-net in base 8, because

1 times m-reduction [i] would yield (60, 170, 1851)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3366 970433 873518 989538 472733 172820 487391 236349 489443 893902 034532 607481 317328 413504 717047 203477 050736 400077 721216 583039 828600 759973 113830 945586 323087 683848 > 8¹⁷⁰ [i]