MinT - Best Known (138, 138+51, s)-Nets in Base 3

Best Known (138, 138+51, s)-Nets in Base 3

(138, 138+51, 288)-Net over F₃ — Constructive and digital

Digital (138, 189, 288)-net over F₃, using

t-expansion [i] based on digital (137, 189, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 63, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(138, 138+51, 607)-Net over F₃ — Digital

Digital (138, 189, 607)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+51, 19679)-Net in Base 3 — Upper bound on s

There is no (138, 189, 19680)-net in base 3, because

1 times m-reduction [i] would yield (138, 188, 19680)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 500302 505337 566829 229082 159011 492654 872594 338681 278308 002956 388215 414884 520900 192432 573377 > 3¹⁸⁸ [i]