MinT - Best Known (205−99, 205, s)-Nets in Base 3

Best Known (205−99, 205, s)-Nets in Base 3

(205−99, 205, 74)-Net over F₃ — Constructive and digital

Digital (106, 205, 74)-net over F₃, using

5 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)

(205−99, 205, 113)-Net over F₃ — Digital

Digital (106, 205, 113)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−99, 205, 878)-Net in Base 3 — Upper bound on s

There is no (106, 205, 879)-net in base 3, because

1 times m-reduction [i] would yield (106, 204, 879)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21 933972 688024 638550 727914 261179 339185 629941 589535 212405 341537 957865 389535 535878 036668 829864 167039 > 3²⁰⁴ [i]