MinT - Best Known (109−49, 109, s)-Nets in Base 3

Best Known (109−49, 109, s)-Nets in Base 3

(109−49, 109, 64)-Net over F₃ — Constructive and digital

Digital (60, 109, 64)-net over F₃, using

1 times m-reduction [i] based on digital (60, 110, 64)-net over F₃, using
- trace code for nets [i] based on digital (5, 55, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]

(109−49, 109, 81)-Net over F₃ — Digital

Digital (60, 109, 81)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−49, 109, 664)-Net in Base 3 — Upper bound on s

There is no (60, 109, 665)-net in base 3, because

1 times m-reduction [i] would yield (60, 108, 665)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3441 287410 616996 373274 386340 116922 310578 987443 781089 > 3¹⁰⁸ [i]