MinT - Best Known (135−77, 135, s)-Nets in Base 8

Best Known (135−77, 135, s)-Nets in Base 8

(135−77, 135, 98)-Net over F₈ — Constructive and digital

Digital (58, 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]

(135−77, 135, 144)-Net over F₈ — Digital

Digital (58, 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]

(135−77, 135, 3259)-Net in Base 8 — Upper bound on s

There is no (58, 135, 3260)-net in base 8, because

1 times m-reduction [i] would yield (58, 134, 3260)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 420748 063118 627329 036915 949794 687056 242865 819521 867763 026296 536851 632652 801510 362354 920223 929916 470509 866907 165815 992678 > 8¹³⁴ [i]