MinT - Best Known (245−65, 245, s)-Nets in Base 3

Best Known (245−65, 245, s)-Nets in Base 3

(245−65, 245, 288)-Net over F₃ — Constructive and digital

Digital (180, 245, 288)-net over F₃, using

t-expansion [i] based on digital (177, 245, 288)-net over F₃, using
- 4 times m-reduction [i] based on digital (177, 249, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 83, 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]

(245−65, 245, 804)-Net over F₃ — Digital

Digital (180, 245, 804)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−65, 245, 27759)-Net in Base 3 — Upper bound on s

There is no (180, 245, 27760)-net in base 3, because

1 times m-reduction [i] would yield (180, 244, 27760)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 261 796759 116874 673868 813387 610385 745696 780887 017708 618687 389534 997191 149931 263203 770399 438343 726966 076963 192602 241025 > 3²⁴⁴ [i]