MinT - Best Known (130−29, 130, s)-Nets in Base 5

Best Known (130−29, 130, s)-Nets in Base 5

(130−29, 130, 504)-Net over F₅ — Constructive and digital

Digital (101, 130, 504)-net over F₅, using

2 times m-reduction [i] based on digital (101, 132, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (35, 50, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 25, 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 (51, 82, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 41, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(130−29, 130, 4982)-Net over F₅ — Digital

Digital (101, 130, 4982)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(130−29, 130, 4167791)-Net in Base 5 — Upper bound on s

There is no (101, 130, 4167792)-net in base 5, because

1 times m-reduction [i] would yield (101, 129, 4167792)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 469368 173259 678422 843842 515082 603675 980021 462169 549852 447737 737170 859857 379094 906813 823745 > 5¹²⁹ [i]