MinT - Best Known (231−110, 231, s)-Nets in Base 3

Best Known (231−110, 231, s)-Nets in Base 3

(231−110, 231, 80)-Net over F₃ — Constructive and digital

Digital (121, 231, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 76, 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]
2. digital (45, 155, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(231−110, 231, 130)-Net over F₃ — Digital

Digital (121, 231, 130)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−110, 231, 1023)-Net in Base 3 — Upper bound on s

There is no (121, 231, 1024)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 171 117381 708320 644439 237691 037830 405651 355988 901967 118461 659867 018813 949585 678774 049980 982326 105869 832618 323969 > 3²³¹ [i]