MinT - Best Known (167−52, 167, s)-Nets in Base 3

Best Known (167−52, 167, s)-Nets in Base 3

(167−52, 167, 158)-Net over F₃ — Constructive and digital

Digital (115, 167, 158)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 29, 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 (86, 138, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 69, 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]

(167−52, 167, 337)-Net over F₃ — Digital

Digital (115, 167, 337)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−52, 167, 6095)-Net in Base 3 — Upper bound on s

There is no (115, 167, 6096)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47 795094 686583 726438 641984 220598 201304 552158 421985 756365 662114 718492 104910 906337 > 3¹⁶⁷ [i]