MinT - Best Known (124−27, 124, s)-Nets in Base 5

Best Known (124−27, 124, s)-Nets in Base 5

(124−27, 124, 504)-Net over F₅ — Constructive and digital

Digital (97, 124, 504)-net over F₅, using

2 times m-reduction [i] based on digital (97, 126, 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 (49, 78, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 39, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(124−27, 124, 5698)-Net over F₅ — Digital

Digital (97, 124, 5698)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(124−27, 124, 5816705)-Net in Base 5 — Upper bound on s

There is no (97, 124, 5816706)-net in base 5, because

1 times m-reduction [i] would yield (97, 123, 5816706)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 94 039654 002991 724044 478810 862256 448657 221441 652260 588009 026478 870873 249083 042236 514825 > 5¹²³ [i]