MinT - Best Known (167, 167+79, s)-Nets in Base 3

Best Known (167, 167+79, s)-Nets in Base 3

(167, 167+79, 176)-Net over F₃ — Constructive and digital

Digital (167, 246, 176)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 54, 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 (113, 192, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 96, 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, 167+79, 428)-Net over F₃ — Digital

Digital (167, 246, 428)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(167, 167+79, 7612)-Net in Base 3 — Upper bound on s

There is no (167, 246, 7613)-net in base 3, because

1 times m-reduction [i] would yield (167, 245, 7613)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 787 993951 753833 589886 127525 905228 924634 850200 163488 347317 285058 384884 395421 554231 745834 328758 327404 160550 268628 239163 > 3²⁴⁵ [i]