MinT - Best Known (209−67, 209, s)-Nets in Base 3

Best Known (209−67, 209, s)-Nets in Base 3

(209−67, 209, 164)-Net over F₃ — Constructive and digital

Digital (142, 209, 164)-net over F₃, using

3¹ times duplication [i] based on digital (141, 208, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 40, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (101, 168, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 84, 74)-net over F₉, using
      - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(209−67, 209, 374)-Net over F₃ — Digital

Digital (142, 209, 374)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−67, 209, 6660)-Net in Base 3 — Upper bound on s

There is no (142, 209, 6661)-net in base 3, because

1 times m-reduction [i] would yield (142, 208, 6661)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1744 701830 999457 240552 926025 671140 430748 234273 400950 490628 492546 849249 946011 254672 763432 855038 205835 > 3²⁰⁸ [i]