MinT - Best Known (54, 54+107, s)-Nets in Base 8

Best Known (54, 54+107, s)-Nets in Base 8

(54, 54+107, 98)-Net over F₈ — Constructive and digital

Digital (54, 161, 98)-net over F₈, using

t-expansion [i] based on digital (37, 161, 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]

(54, 54+107, 144)-Net over F₈ — Digital

Digital (54, 161, 144)-net over F₈, using

t-expansion [i] based on digital (45, 161, 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]

(54, 54+107, 1533)-Net in Base 8 — Upper bound on s

There is no (54, 161, 1534)-net in base 8, because

1 times m-reduction [i] would yield (54, 160, 1534)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 185265 615935 335627 239297 922499 724026 336860 054939 279104 451718 929181 208198 560378 937205 642894 382851 302561 785115 283189 173663 158968 531729 704468 644525 > 8¹⁶⁰ [i]