MinT - Best Known (138−67, 138, s)-Nets in Base 3

Best Known (138−67, 138, s)-Nets in Base 3

(138−67, 138, 60)-Net over F₃ — Constructive and digital

Digital (71, 138, 60)-net over F₃, using

3 times m-reduction [i] based on digital (71, 141, 60)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 50, 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 (21, 91, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]

(138−67, 138, 84)-Net over F₃ — Digital

Digital (71, 138, 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]

(138−67, 138, 597)-Net in Base 3 — Upper bound on s

There is no (71, 138, 598)-net in base 3, because

1 times m-reduction [i] would yield (71, 137, 598)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 234111 312129 881726 969314 760309 798480 721391 316335 648056 556241 423469 > 3¹³⁷ [i]