MinT - Best Known (207−91, 207, s)-Nets in Base 3

Best Known (207−91, 207, s)-Nets in Base 3

(207−91, 207, 84)-Net over F₃ — Constructive and digital

Digital (116, 207, 84)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 71, 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]
2. digital (45, 136, 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]

(207−91, 207, 148)-Net over F₃ — Digital

Digital (116, 207, 148)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−91, 207, 1302)-Net in Base 3 — Upper bound on s

There is no (116, 207, 1303)-net in base 3, because

1 times m-reduction [i] would yield (116, 206, 1303)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 194 353473 810277 162220 269653 169969 088953 604051 919229 525161 825318 048177 235211 567465 867336 000406 611607 > 3²⁰⁶ [i]