MinT - Best Known (216−96, 216, s)-Nets in Base 3

Best Known (216−96, 216, s)-Nets in Base 3

(216−96, 216, 85)-Net over F₃ — Constructive and digital

Digital (120, 216, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 75, 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]
2. digital (45, 141, 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]

(216−96, 216, 149)-Net over F₃ — Digital

Digital (120, 216, 149)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−96, 216, 1267)-Net in Base 3 — Upper bound on s

There is no (120, 216, 1268)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 501796 390085 869096 254173 587977 568973 351159 195919 217632 664496 469040 542380 230194 423837 438462 244658 653569 > 3²¹⁶ [i]