MinT - Best Known (88, 88+82, s)-Nets in Base 3

Best Known (88, 88+82, s)-Nets in Base 3

(88, 88+82, 68)-Net over F₃ — Constructive and digital

Digital (88, 170, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 62, 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 (26, 108, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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+82, 96)-Net over F₃ — Digital

Digital (88, 170, 96)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(88, 88+82, 728)-Net in Base 3 — Upper bound on s

There is no (88, 170, 729)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1356 178553 296188 110716 783236 292538 590753 683980 523977 987946 773080 621719 338989 590163 > 3¹⁷⁰ [i]