MinT - Best Known (121, 121+107, s)-Nets in Base 3

Best Known (121, 121+107, s)-Nets in Base 3

(121, 121+107, 80)-Net over F₃ — Constructive and digital

Digital (121, 228, 80)-net over F₃, using

3 times m-reduction [i] based on 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]

(121, 121+107, 134)-Net over F₃ — Digital

Digital (121, 228, 134)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(121, 121+107, 1086)-Net in Base 3 — Upper bound on s

There is no (121, 228, 1087)-net in base 3, because

1 times m-reduction [i] would yield (121, 227, 1087)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 039988 542666 537306 843113 660388 867028 648918 810447 476523 750437 356786 706363 848605 159387 360734 065713 388767 860247 > 3²²⁷ [i]