MinT - Best Known (89, 89+57, s)-Nets in Base 5

Best Known (89, 89+57, s)-Nets in Base 5

(89, 89+57, 252)-Net over F₅ — Constructive and digital

Digital (89, 146, 252)-net over F₅, using

t-expansion [i] based on digital (85, 146, 252)-net over F₅, using
- 4 times m-reduction [i] based on digital (85, 150, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 75, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(89, 89+57, 341)-Net over F₅ — Digital

Digital (89, 146, 341)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+57, 11745)-Net in Base 5 — Upper bound on s

There is no (89, 146, 11746)-net in base 5, because

1 times m-reduction [i] would yield (89, 145, 11746)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 224724 323555 932149 331515 588210 058889 666952 794701 741276 320684 878687 176021 950578 419668 673180 707779 707585 > 5¹⁴⁵ [i]