MinT - Best Known (130, 130+109, s)-Nets in Base 3

Best Known (130, 130+109, s)-Nets in Base 3

(130, 130+109, 85)-Net over F₃ — Constructive and digital

Digital (130, 239, 85)-net over F₃, using

7 times m-reduction [i] based on digital (130, 246, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 85, 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]
  2. digital (45, 161, 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]

(130, 130+109, 152)-Net over F₃ — Digital

Digital (130, 239, 152)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(130, 130+109, 1275)-Net in Base 3 — Upper bound on s

There is no (130, 239, 1276)-net in base 3, because

1 times m-reduction [i] would yield (130, 238, 1276)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 359718 586965 522517 676587 667390 186480 228294 922871 404039 876305 681006 914604 552772 506813 456166 361747 075090 967907 539881 > 3²³⁸ [i]