MinT - Best Known (139−69, 139, s)-Nets in Base 3

Best Known (139−69, 139, s)-Nets in Base 3

(139−69, 139, 60)-Net over F₃ — Constructive and digital

Digital (70, 139, 60)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 49, 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, 90, 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]

(139−69, 139, 82)-Net over F₃ — Digital

Digital (70, 139, 82)-net over F₃, using

t-expansion [i] based on digital (69, 139, 82)-net over F₃, using
- net from sequence [i] based on digital (69, 81)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
    - Drinfeld modules of rank 1 [i]

(139−69, 139, 552)-Net in Base 3 — Upper bound on s

There is no (70, 139, 553)-net in base 3, because

1 times m-reduction [i] would yield (70, 138, 553)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 735233 321428 106586 132520 611649 602798 783333 921633 679042 854354 633369 > 3¹³⁸ [i]