MinT - Best Known (61, 61+57, s)-Nets in Base 3

Best Known (61, 61+57, s)-Nets in Base 3

(61, 61+57, 56)-Net over F₃ — Constructive and digital

Digital (61, 118, 56)-net over F₃, using

5 times m-reduction [i] based on digital (61, 123, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 46, 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, 77, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(61, 61+57, 71)-Net over F₃ — Digital

Digital (61, 118, 71)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+57, 529)-Net in Base 3 — Upper bound on s

There is no (61, 118, 530)-net in base 3, because

1 times m-reduction [i] would yield (61, 117, 530)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 67 220051 782798 639636 769083 128037 391252 706098 012238 877465 > 3¹¹⁷ [i]