MinT - Best Known (229−75, 229, s)-Nets in Base 3

Best Known (229−75, 229, s)-Nets in Base 3

(229−75, 229, 164)-Net over F₃ — Constructive and digital

Digital (154, 229, 164)-net over F₃, using

3¹ times duplication [i] based on digital (153, 228, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 44, 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 (109, 184, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 92, 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]

(229−75, 229, 378)-Net over F₃ — Digital

Digital (154, 229, 378)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−75, 229, 6345)-Net in Base 3 — Upper bound on s

There is no (154, 229, 6346)-net in base 3, because

1 times m-reduction [i] would yield (154, 228, 6346)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 078754 850783 243674 109059 869260 229570 198147 124470 864984 328037 250014 457010 215746 428989 695919 871179 076997 448621 > 3²²⁸ [i]