MinT - Best Known (70, 131, s)-Nets in Base 3

Best Known (70, 131, s)-Nets in Base 3

(70, 131, 61)-Net over F₃ — Constructive and digital

Digital (70, 131, 61)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 43, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (27, 88, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(70, 131, 86)-Net over F₃ — Digital

Digital (70, 131, 86)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 131, 674)-Net in Base 3 — Upper bound on s

There is no (70, 131, 675)-net in base 3, because

1 times m-reduction [i] would yield (70, 130, 675)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 108 225676 520402 918887 487407 654785 177996 622566 390291 441065 062125 > 3¹³⁰ [i]