MinT - Best Known (126, 126+118, s)-Nets in Base 3

Best Known (126, 126+118, s)-Nets in Base 3

(126, 126+118, 80)-Net over F₃ — Constructive and digital

Digital (126, 244, 80)-net over F₃, using

2 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+118, 131)-Net over F₃ — Digital

Digital (126, 244, 131)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+118, 1015)-Net in Base 3 — Upper bound on s

There is no (126, 244, 1016)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 268 613929 353936 484697 501673 601501 217793 098914 919320 686988 589518 249302 778483 632116 132230 293130 132051 957565 656352 540001 > 3²⁴⁴ [i]