MinT - Best Known (138−63, 138, s)-Nets in Base 5

Best Known (138−63, 138, s)-Nets in Base 5

(138−63, 138, 132)-Net over F₅ — Constructive and digital

Digital (75, 138, 132)-net over F₅, using

4 times m-reduction [i] based on digital (75, 142, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 71, 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]

(138−63, 138, 185)-Net over F₅ — Digital

Digital (75, 138, 185)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−63, 138, 3787)-Net in Base 5 — Upper bound on s

There is no (75, 138, 3788)-net in base 5, because

1 times m-reduction [i] would yield (75, 137, 3788)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 575427 845113 179862 954769 511274 341865 351449 268956 831953 633389 797613 740901 958579 625085 963066 047921 > 5¹³⁷ [i]