MinT - Best Known (109−61, 109, s)-Nets in Base 8

Best Known (109−61, 109, s)-Nets in Base 8

(109−61, 109, 98)-Net over F₈ — Constructive and digital

Digital (48, 109, 98)-net over F₈, using

t-expansion [i] based on digital (37, 109, 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]

(109−61, 109, 144)-Net over F₈ — Digital

Digital (48, 109, 144)-net over F₈, using

t-expansion [i] based on digital (45, 109, 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]

(109−61, 109, 3048)-Net in Base 8 — Upper bound on s

There is no (48, 109, 3049)-net in base 8, because

1 times m-reduction [i] would yield (48, 108, 3049)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 188649 967993 027056 385725 846528 958027 289302 149530 765436 395878 206504 414005 869311 607754 452004 127440 > 8¹⁰⁸ [i]