MinT - Best Known (78, 78+29, s)-Nets in Base 5

Best Known (78, 78+29, s)-Nets in Base 5

(78, 78+29, 304)-Net over F₅ — Constructive and digital

Digital (78, 107, 304)-net over F₅, using

5¹ times duplication [i] based on 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]

(78, 78+29, 1338)-Net over F₅ — Digital

Digital (78, 107, 1338)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+29, 296199)-Net in Base 5 — Upper bound on s

There is no (78, 107, 296200)-net in base 5, because

1 times m-reduction [i] would yield (78, 106, 296200)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 123 261528 810187 665424 129483 033239 374422 220158 794630 852131 544127 772873 613441 > 5¹⁰⁶ [i]