MinT - Best Known (129−66, 129, s)-Nets in Base 3

Best Known (129−66, 129, s)-Nets in Base 3

(129−66, 129, 56)-Net over F₃ — Constructive and digital

Digital (63, 129, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 48, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 81, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(129−66, 129, 65)-Net over F₃ — Digital

Digital (63, 129, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−66, 129, 450)-Net in Base 3 — Upper bound on s

There is no (63, 129, 451)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35 829128 198082 169703 757518 036803 280679 885106 883472 532633 074759 > 3¹²⁹ [i]