MinT - Best Known (123−27, 123, s)-Nets in Base 5

Best Known (123−27, 123, s)-Nets in Base 5

(123−27, 123, 504)-Net over F₅ — Constructive and digital

Digital (96, 123, 504)-net over F₅, using

1 times m-reduction [i] based on digital (96, 124, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (34, 48, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 24, 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]
  2. digital (48, 76, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 38, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(123−27, 123, 5357)-Net over F₅ — Digital

Digital (96, 123, 5357)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−27, 123, 5139372)-Net in Base 5 — Upper bound on s

There is no (96, 123, 5139373)-net in base 5, because

1 times m-reduction [i] would yield (96, 122, 5139373)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 18 807953 143120 372812 330169 658872 894984 249510 290132 179710 897233 545084 889967 469377 848645 > 5¹²² [i]