MinT - Best Known (78, 78+61, s)-Nets in Base 5

Best Known (78, 78+61, s)-Nets in Base 5

(78, 78+61, 132)-Net over F₅ — Constructive and digital

Digital (78, 139, 132)-net over F₅, using

9 times m-reduction [i] based on digital (78, 148, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 74, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(78, 78+61, 214)-Net over F₅ — Digital

Digital (78, 139, 214)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+61, 4921)-Net in Base 5 — Upper bound on s

There is no (78, 139, 4922)-net in base 5, because

1 times m-reduction [i] would yield (78, 138, 4922)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 886697 699847 760835 848525 469697 883562 532613 446376 511886 178133 299138 017532 987897 520335 056402 270209 > 5¹³⁸ [i]