MinT - Best Known (206−103, 206, s)-Nets in Base 3

Best Known (206−103, 206, s)-Nets in Base 3

(206−103, 206, 72)-Net over F₃ — Constructive and digital

Digital (103, 206, 72)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 77, 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, 129, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃ (see above)

(206−103, 206, 104)-Net over F₃ — Digital

Digital (103, 206, 104)-net over F₃, using

t-expansion [i] based on digital (102, 206, 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]

(206−103, 206, 772)-Net in Base 3 — Upper bound on s

There is no (103, 206, 773)-net in base 3, because

1 times m-reduction [i] would yield (103, 205, 773)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 67 150882 358128 791360 135562 262803 565376 208076 077158 453138 260888 153715 493370 622832 075804 864212 218283 > 3²⁰⁵ [i]