MinT - Best Known (207−101, 207, s)-Nets in Base 3

Best Known (207−101, 207, s)-Nets in Base 3

(207−101, 207, 74)-Net over F₃ — Constructive and digital

Digital (106, 207, 74)-net over F₃, using

3 times m-reduction [i] based on digital (106, 210, 74)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 79, 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 (27, 131, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(207−101, 207, 110)-Net over F₃ — Digital

Digital (106, 207, 110)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−101, 207, 851)-Net in Base 3 — Upper bound on s

There is no (106, 207, 852)-net in base 3, because

1 times m-reduction [i] would yield (106, 206, 852)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 195 056355 575817 082206 045317 739084 643950 051906 045587 110282 559890 926625 842888 483116 456427 787024 729881 > 3²⁰⁶ [i]