MinT - Best Known (54, 54+69, s)-Nets in Base 8

Best Known (54, 54+69, s)-Nets in Base 8

(54, 54+69, 98)-Net over F₈ — Constructive and digital

Digital (54, 123, 98)-net over F₈, using

t-expansion [i] based on digital (37, 123, 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+69, 144)-Net over F₈ — Digital

Digital (54, 123, 144)-net over F₈, using

t-expansion [i] based on digital (45, 123, 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+69, 3342)-Net in Base 8 — Upper bound on s

There is no (54, 123, 3343)-net in base 8, because

1 times m-reduction [i] would yield (54, 122, 3343)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 542145 082777 421005 482806 377848 202698 540192 543832 644462 428499 167383 321792 371936 283742 764605 383484 627612 575144 > 8¹²² [i]