MinT - Best Known (135−61, 135, s)-Nets in Base 5

Best Known (135−61, 135, s)-Nets in Base 5

(135−61, 135, 132)-Net over F₅ — Constructive and digital

Digital (74, 135, 132)-net over F₅, using

5 times m-reduction [i] based on digital (74, 140, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 70, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(135−61, 135, 188)-Net over F₅ — Digital

Digital (74, 135, 188)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−61, 135, 3966)-Net in Base 5 — Upper bound on s

There is no (74, 135, 3967)-net in base 5, because

1 times m-reduction [i] would yield (74, 134, 3967)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4612 317168 944263 545408 263381 644166 772173 761453 073081 651128 968702 602213 012417 725351 193424 211849 > 5¹³⁴ [i]