MinT - Best Known (88, 88+78, s)-Nets in Base 3

Best Known (88, 88+78, s)-Nets in Base 3

(88, 88+78, 69)-Net over F₃ — Constructive and digital

Digital (88, 166, 69)-net over F₃, using

2 times m-reduction [i] based on digital (88, 168, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 61, 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, 107, 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]

(88, 88+78, 102)-Net over F₃ — Digital

Digital (88, 166, 102)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(88, 88+78, 788)-Net in Base 3 — Upper bound on s

There is no (88, 166, 789)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16 165767 809616 063349 734418 205349 949373 307955 645268 174175 425180 073729 007635 873435 > 3¹⁶⁶ [i]