MinT - Best Known (135−81, 135, s)-Nets in Base 8

Best Known (135−81, 135, s)-Nets in Base 8

(135−81, 135, 98)-Net over F₈ — Constructive and digital

Digital (54, 135, 98)-net over F₈, using

t-expansion [i] based on digital (37, 135, 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]

(135−81, 135, 144)-Net over F₈ — Digital

Digital (54, 135, 144)-net over F₈, using

t-expansion [i] based on digital (45, 135, 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]

(135−81, 135, 2362)-Net in Base 8 — Upper bound on s

There is no (54, 135, 2363)-net in base 8, because

1 times m-reduction [i] would yield (54, 134, 2363)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 342368 377017 509548 195551 407007 439627 193951 098765 264240 907583 936271 052090 687477 000595 317697 854102 739488 407860 158074 612061 > 8¹³⁴ [i]