MinT - Best Known (54, 54+27, s)-Nets in Base 5

Best Known (54, 54+27, s)-Nets in Base 5

(54, 54+27, 252)-Net over F₅ — Constructive and digital

Digital (54, 81, 252)-net over F₅, using

7 times m-reduction [i] based on digital (54, 88, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 44, 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]

(54, 54+27, 410)-Net over F₅ — Digital

Digital (54, 81, 410)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+27, 28350)-Net in Base 5 — Upper bound on s

There is no (54, 81, 28351)-net in base 5, because

1 times m-reduction [i] would yield (54, 80, 28351)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 82 742771 638117 205075 602490 963936 760102 129489 314132 830925 > 5⁸⁰ [i]