MinT - Best Known (166−109, 166, s)-Nets in Base 8

Best Known (166−109, 166, s)-Nets in Base 8

(166−109, 166, 98)-Net over F₈ — Constructive and digital

Digital (57, 166, 98)-net over F₈, using

t-expansion [i] based on digital (37, 166, 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]

(166−109, 166, 144)-Net over F₈ — Digital

Digital (57, 166, 144)-net over F₈, using

t-expansion [i] based on digital (45, 166, 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]

(166−109, 166, 1687)-Net in Base 8 — Upper bound on s

There is no (57, 166, 1688)-net in base 8, because

1 times m-reduction [i] would yield (57, 165, 1688)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 104207 648756 591951 270957 770026 480029 650743 804006 173552 037653 157126 216549 605507 471198 537629 708000 795666 225277 421487 992802 279092 339972 231163 695632 858188 > 8¹⁶⁵ [i]