MinT - Best Known (126, 126+115, s)-Nets in Base 3

Best Known (126, 126+115, s)-Nets in Base 3

(126, 126+115, 80)-Net over F₃ — Constructive and digital

Digital (126, 241, 80)-net over F₃, using

5 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]

(126, 126+115, 134)-Net over F₃ — Digital

Digital (126, 241, 134)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+115, 1071)-Net in Base 3 — Upper bound on s

There is no (126, 241, 1072)-net in base 3, because

1 times m-reduction [i] would yield (126, 240, 1072)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 298675 701639 226667 650187 341404 889278 113274 544436 742614 835816 336043 809072 602420 275256 951140 967247 636307 019414 631777 > 3²⁴⁰ [i]