MinT - Best Known (45, 45+61, s)-Nets in Base 8

Best Known (45, 45+61, s)-Nets in Base 8

(45, 45+61, 98)-Net over F₈ — Constructive and digital

Digital (45, 106, 98)-net over F₈, using

t-expansion [i] based on digital (37, 106, 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]

(45, 45+61, 144)-Net over F₈ — Digital

Digital (45, 106, 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]

(45, 45+61, 2473)-Net in Base 8 — Upper bound on s

There is no (45, 106, 2474)-net in base 8, because

1 times m-reduction [i] would yield (45, 105, 2474)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 67546 491946 617192 686890 420225 060997 074825 580832 501999 668031 047231 431316 027170 735514 711681 178384 > 8¹⁰⁵ [i]