MinT - Best Known (42, 82, s)-Nets in Base 3

Best Known (42, 82, s)-Nets in Base 3

(42, 82, 44)-Net over F₃ — Constructive and digital

Digital (42, 82, 44)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 27, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 55, 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]

(42, 82, 56)-Net over F₃ — Digital

Digital (42, 82, 56)-net over F₃, using

t-expansion [i] based on digital (40, 82, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(42, 82, 356)-Net in Base 3 — Upper bound on s

There is no (42, 82, 357)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1381 257764 840728 202259 219045 785010 181137 > 3⁸² [i]