MinT - Best Known (83, 83+70, s)-Nets in Base 3

Best Known (83, 83+70, s)-Nets in Base 3

(83, 83+70, 69)-Net over F₃ — Constructive and digital

Digital (83, 153, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 56, 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 (27, 97, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(83, 83+70, 102)-Net over F₃ — Digital

Digital (83, 153, 102)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+70, 813)-Net in Base 3 — Upper bound on s

There is no (83, 153, 814)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 336134 949102 121923 758589 633344 813060 022699 425997 235117 430973 218452 857089 > 3¹⁵³ [i]