MinT - Best Known (122, 122+111, s)-Nets in Base 3

Best Known (122, 122+111, s)-Nets in Base 3

(122, 122+111, 80)-Net over F₃ — Constructive and digital

Digital (122, 233, 80)-net over F₃, using

1 times m-reduction [i] based on digital (122, 234, 80)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 77, 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, 157, 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]

(122, 122+111, 131)-Net over F₃ — Digital

Digital (122, 233, 131)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(122, 122+111, 1044)-Net in Base 3 — Upper bound on s

There is no (122, 233, 1045)-net in base 3, because

1 times m-reduction [i] would yield (122, 232, 1045)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 495 279369 871829 867998 941984 158785 319695 541863 334301 253536 975019 312984 019297 536976 048830 555520 826685 921956 019419 > 3²³² [i]