MinT - Best Known (209−105, 209, s)-Nets in Base 3

Best Known (209−105, 209, s)-Nets in Base 3

(209−105, 209, 72)-Net over F₃ — Constructive and digital

Digital (104, 209, 72)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 78, 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 (26, 131, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃ (see above)

(209−105, 209, 104)-Net over F₃ — Digital

Digital (104, 209, 104)-net over F₃, using

t-expansion [i] based on digital (102, 209, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(209−105, 209, 768)-Net in Base 3 — Upper bound on s

There is no (104, 209, 769)-net in base 3, because

1 times m-reduction [i] would yield (104, 208, 769)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1750 055199 177221 079571 218108 668383 042124 129280 015041 628719 816367 980598 480209 222883 322635 809379 621393 > 3²⁰⁸ [i]