MinT - Best Known (164−51, 164, s)-Nets in Base 3

Best Known (164−51, 164, s)-Nets in Base 3

(164−51, 164, 158)-Net over F₃ — Constructive and digital

Digital (113, 164, 158)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 28, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 136, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 68, 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]

(164−51, 164, 333)-Net over F₃ — Digital

Digital (113, 164, 333)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−51, 164, 6543)-Net in Base 3 — Upper bound on s

There is no (113, 164, 6544)-net in base 3, because

1 times m-reduction [i] would yield (113, 163, 6544)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 591148 084525 259566 027651 555636 497588 633682 976537 277133 247223 260701 951173 453345 > 3¹⁶³ [i]