MinT - Best Known (69, 69+43, s)-Nets in Base 5

Best Known (69, 69+43, s)-Nets in Base 5

(69, 69+43, 252)-Net over F₅ — Constructive and digital

Digital (69, 112, 252)-net over F₅, using

6 times m-reduction [i] based on digital (69, 118, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 59, 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]

(69, 69+43, 292)-Net over F₅ — Digital

Digital (69, 112, 292)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+43, 10724)-Net in Base 5 — Upper bound on s

There is no (69, 112, 10725)-net in base 5, because

1 times m-reduction [i] would yield (69, 111, 10725)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 385666 106809 018326 859051 105562 694318 700339 884417 931223 379350 510121 951110 336901 > 5¹¹¹ [i]