MinT - Best Known (137−71, 137, s)-Nets in Base 3

Best Known (137−71, 137, s)-Nets in Base 3

(137−71, 137, 56)-Net over F₃ — Constructive and digital

Digital (66, 137, 56)-net over F₃, using

1 times m-reduction [i] based on digital (66, 138, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 51, 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, 87, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(137−71, 137, 66)-Net over F₃ — Digital

Digital (66, 137, 66)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−71, 137, 463)-Net in Base 3 — Upper bound on s

There is no (66, 137, 464)-net in base 3, because

1 times m-reduction [i] would yield (66, 136, 464)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 81419 509671 727706 723148 388959 082934 098879 321401 146075 253555 482433 > 3¹³⁶ [i]