MinT - Best Known (207−97, 207, s)-Nets in Base 3

Best Known (207−97, 207, s)-Nets in Base 3

(207−97, 207, 76)-Net over F₃ — Constructive and digital

Digital (110, 207, 76)-net over F₃, using

3 times m-reduction [i] based on digital (110, 210, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 65, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 145, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(207−97, 207, 124)-Net over F₃ — Digital

Digital (110, 207, 124)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−97, 207, 999)-Net in Base 3 — Upper bound on s

There is no (110, 207, 1000)-net in base 3, because

1 times m-reduction [i] would yield (110, 206, 1000)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 202 049442 644297 292429 491678 243741 439371 758289 298498 418501 008515 771567 009881 119002 654269 691172 654081 > 3²⁰⁶ [i]