MinT - Best Known (57, 110, s)-Nets in Base 3

Best Known (57, 110, s)-Nets in Base 3

(57, 110, 56)-Net over F₃ — Constructive and digital

Digital (57, 110, 56)-net over F₃, using

1 times m-reduction [i] based on digital (57, 111, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 42, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 69, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(57, 110, 68)-Net over F₃ — Digital

Digital (57, 110, 68)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 110, 502)-Net in Base 3 — Upper bound on s

There is no (57, 110, 503)-net in base 3, because

1 times m-reduction [i] would yield (57, 109, 503)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10223 742009 327882 827346 355008 350268 243490 222725 029181 > 3¹⁰⁹ [i]