MinT - Best Known (145−74, 145, s)-Nets in Base 3

Best Known (145−74, 145, s)-Nets in Base 3

(145−74, 145, 56)-Net over F₃ — Constructive and digital

Digital (71, 145, 56)-net over F₃, using

8 times m-reduction [i] based on digital (71, 153, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 56, 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, 97, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(145−74, 145, 84)-Net over F₃ — Digital

Digital (71, 145, 84)-net over F₃, using

net from sequence [i] based on digital (71, 83)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(145−74, 145, 507)-Net in Base 3 — Upper bound on s

There is no (71, 145, 508)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1592 740472 552133 833648 704577 058852 937104 353145 312532 983104 686440 691161 > 3¹⁴⁵ [i]