MinT - Best Known (137−81, 137, s)-Nets in Base 8

Best Known (137−81, 137, s)-Nets in Base 8

(137−81, 137, 98)-Net over F₈ — Constructive and digital

Digital (56, 137, 98)-net over F₈, using

t-expansion [i] based on digital (37, 137, 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]

(137−81, 137, 144)-Net over F₈ — Digital

Digital (56, 137, 144)-net over F₈, using

t-expansion [i] based on digital (45, 137, 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]

(137−81, 137, 2624)-Net in Base 8 — Upper bound on s

There is no (56, 137, 2625)-net in base 8, because

1 times m-reduction [i] would yield (56, 136, 2625)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 665 331831 441420 602574 268706 569016 402530 508942 573892 537400 663074 906150 711320 550770 190163 585798 987757 960154 643333 556154 608136 > 8¹³⁶ [i]