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

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

(106−29, 106, 304)-Net over F₅ — Constructive and digital

Digital (77, 106, 304)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (14, 28, 52)-net over F₅, using
  - trace code for nets [i] based on digital (0, 14, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
      - Niederreiter sequence [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, 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]

(106−29, 106, 1264)-Net over F₅ — Digital

Digital (77, 106, 1264)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−29, 106, 264031)-Net in Base 5 — Upper bound on s

There is no (77, 106, 264032)-net in base 5, because

1 times m-reduction [i] would yield (77, 105, 264032)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 24 652088 311056 144353 536667 856190 574304 406191 647098 270306 031295 787361 857025 > 5¹⁰⁵ [i]