MinT - Best Known (136−31, 136, s)-Nets in Base 5

Best Known (136−31, 136, s)-Nets in Base 5

(136−31, 136, 504)-Net over F₅ — Constructive and digital

Digital (105, 136, 504)-net over F₅, using

2 times m-reduction [i] based on digital (105, 138, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (36, 52, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 26, 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 (53, 86, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 43, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(136−31, 136, 4455)-Net over F₅ — Digital

Digital (105, 136, 4455)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−31, 136, 3136427)-Net in Base 5 — Upper bound on s

There is no (105, 136, 3136428)-net in base 5, because

1 times m-reduction [i] would yield (105, 135, 3136428)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 22958 971090 689867 818061 236859 003349 251738 799892 185690 590343 286239 419726 453645 178407 450181 107505 > 5¹³⁵ [i]