MinT - Best Known (144−99, 144, s)-Nets in Base 8

Best Known (144−99, 144, s)-Nets in Base 8

(144−99, 144, 98)-Net over F₈ — Constructive and digital

Digital (45, 144, 98)-net over F₈, using

t-expansion [i] based on digital (37, 144, 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]

(144−99, 144, 144)-Net over F₈ — Digital

Digital (45, 144, 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]

(144−99, 144, 1148)-Net in Base 8 — Upper bound on s

There is no (45, 144, 1149)-net in base 8, because

1 times m-reduction [i] would yield (45, 143, 1149)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1391 561199 841622 716715 941109 544474 752043 937531 600701 330812 905867 657382 725801 580973 548322 685691 409100 117362 048090 643299 776864 052684 > 8¹⁴³ [i]