MinT - Best Known (217−101, 217, s)-Nets in Base 3

Best Known (217−101, 217, s)-Nets in Base 3

(217−101, 217, 80)-Net over F₃ — Constructive and digital

Digital (116, 217, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 71, 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, 146, 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]

(217−101, 217, 131)-Net over F₃ — Digital

Digital (116, 217, 131)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−101, 217, 1072)-Net in Base 3 — Upper bound on s

There is no (116, 217, 1073)-net in base 3, because

1 times m-reduction [i] would yield (116, 216, 1073)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 495471 261143 556729 322538 122805 776184 380192 815605 474820 101830 995487 385737 485673 887201 939820 243802 182057 > 3²¹⁶ [i]