MinT - Best Known (172−47, 172, s)-Nets in Base 3

Best Known (172−47, 172, s)-Nets in Base 3

(172−47, 172, 282)-Net over F₃ — Constructive and digital

Digital (125, 172, 282)-net over F₃, using

3¹ times duplication [i] based on digital (124, 171, 282)-net over F₃, using
- trace code for nets [i] based on digital (10, 57, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(172−47, 172, 536)-Net over F₃ — Digital

Digital (125, 172, 536)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−47, 172, 16600)-Net in Base 3 — Upper bound on s

There is no (125, 172, 16601)-net in base 3, because

1 times m-reduction [i] would yield (125, 171, 16601)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3870 834307 907898 362045 193130 626730 976636 196504 712131 466838 384041 310401 881727 313067 > 3¹⁷¹ [i]