MinT - Best Known (243−117, 243, s)-Nets in Base 3

Best Known (243−117, 243, s)-Nets in Base 3

(243−117, 243, 80)-Net over F₃ — Constructive and digital

Digital (126, 243, 80)-net over F₃, using

3 times m-reduction [i] based on digital (126, 246, 80)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 81, 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, 165, 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]

(243−117, 243, 132)-Net over F₃ — Digital

Digital (126, 243, 132)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−117, 243, 1042)-Net in Base 3 — Upper bound on s

There is no (126, 243, 1043)-net in base 3, because

1 times m-reduction [i] would yield (126, 242, 1043)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 708298 196483 190884 331450 825870 995434 788421 037186 760748 833893 312073 069074 167838 397461 191543 323894 594318 703765 196949 > 3²⁴² [i]