MinT - Best Known (139−67, 139, s)-Nets in Base 3

Best Known (139−67, 139, s)-Nets in Base 3

(139−67, 139, 60)-Net over F₃ — Constructive and digital

Digital (72, 139, 60)-net over F₃, using

5 times m-reduction [i] based on digital (72, 144, 60)-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 (21, 93, 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]

(139−67, 139, 84)-Net over F₃ — Digital

Digital (72, 139, 84)-net over F₃, using

t-expansion [i] based on digital (71, 139, 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]

(139−67, 139, 619)-Net in Base 3 — Upper bound on s

There is no (72, 139, 620)-net in base 3, because

1 times m-reduction [i] would yield (72, 138, 620)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 727757 047355 201812 692954 234648 407448 356934 369205 714286 651706 299609 > 3¹³⁸ [i]