MinT - Best Known (139−85, 139, s)-Nets in Base 8

Best Known (139−85, 139, s)-Nets in Base 8

(139−85, 139, 98)-Net over F₈ — Constructive and digital

Digital (54, 139, 98)-net over F₈, using

t-expansion [i] based on digital (37, 139, 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]

(139−85, 139, 144)-Net over F₈ — Digital

Digital (54, 139, 144)-net over F₈, using

t-expansion [i] based on digital (45, 139, 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]

(139−85, 139, 2161)-Net in Base 8 — Upper bound on s

There is no (54, 139, 2162)-net in base 8, because

1 times m-reduction [i] would yield (54, 138, 2162)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42833 502965 866957 863269 497026 994135 325114 272241 532627 828172 720814 410791 046047 120789 834842 422361 987129 275342 449757 397899 019480 > 8¹³⁸ [i]