MinT - Best Known (106, 106+29, s)-Nets in Base 5

Best Known (106, 106+29, s)-Nets in Base 5

(106, 106+29, 510)-Net over F₅ — Constructive and digital

Digital (106, 135, 510)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 9, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. 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]
3. 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)

(106, 106+29, 6636)-Net over F₅ — Digital

Digital (106, 135, 6636)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(106, 106+29, 7405231)-Net in Base 5 — Upper bound on s

There is no (106, 135, 7405232)-net in base 5, because

1 times m-reduction [i] would yield (106, 134, 7405232)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4591 780352 738863 271998 048522 379130 731520 529261 702132 015861 936236 902792 029238 318297 228833 812225 > 5¹³⁴ [i]