MinT - Best Known (164, 164+59, s)-Nets in Base 3

Best Known (164, 164+59, s)-Nets in Base 3

(164, 164+59, 288)-Net over F₃ — Constructive and digital

Digital (164, 223, 288)-net over F₃, using

t-expansion [i] based on digital (163, 223, 288)-net over F₃, using
- 5 times m-reduction [i] based on digital (163, 228, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 76, 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]

(164, 164+59, 749)-Net over F₃ — Digital

Digital (164, 223, 749)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+59, 26185)-Net in Base 3 — Upper bound on s

There is no (164, 223, 26186)-net in base 3, because

1 times m-reduction [i] would yield (164, 222, 26186)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8337 399590 027143 329574 061413 475335 654234 462676 013675 259923 173403 566685 404665 274551 899409 009663 558219 455357 > 3²²² [i]