MinT - Best Known (215−95, 215, s)-Nets in Base 3

Best Known (215−95, 215, s)-Nets in Base 3

(215−95, 215, 85)-Net over F₃ — Constructive and digital

Digital (120, 215, 85)-net over F₃, using

1 times m-reduction [i] based on digital (120, 216, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 75, 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, 141, 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]

(215−95, 215, 151)-Net over F₃ — Digital

Digital (120, 215, 151)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−95, 215, 1320)-Net in Base 3 — Upper bound on s

There is no (120, 215, 1321)-net in base 3, because

1 times m-reduction [i] would yield (120, 214, 1321)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 302130 285880 473788 895686 267043 911116 197533 061829 321343 000062 129734 174903 491936 643647 766345 688142 163083 > 3²¹⁴ [i]