MinT - Best Known (209−103, 209, s)-Nets in Base 3

Best Known (209−103, 209, s)-Nets in Base 3

(209−103, 209, 74)-Net over F₃ — Constructive and digital

Digital (106, 209, 74)-net over F₃, using

1 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)

(209−103, 209, 108)-Net over F₃ — Digital

Digital (106, 209, 108)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−103, 209, 827)-Net in Base 3 — Upper bound on s

There is no (106, 209, 828)-net in base 3, because

1 times m-reduction [i] would yield (106, 208, 828)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1832 608151 521286 985086 490144 554304 812930 757065 279334 965232 885560 941716 879878 274273 745843 114371 198609 > 3²⁰⁸ [i]