MinT - Best Known (172−51, 172, s)-Nets in Base 3

Best Known (172−51, 172, s)-Nets in Base 3

(172−51, 172, 228)-Net over F₃ — Constructive and digital

Digital (121, 172, 228)-net over F₃, using

3¹ times duplication [i] based on digital (120, 171, 228)-net over F₃, using
- trace code for nets [i] based on digital (6, 57, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(172−51, 172, 404)-Net over F₃ — Digital

Digital (121, 172, 404)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−51, 172, 9310)-Net in Base 3 — Upper bound on s

There is no (121, 172, 9311)-net in base 3, because

1 times m-reduction [i] would yield (121, 171, 9311)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3877 149710 822159 052020 644756 067551 298140 138187 621775 694072 835667 310229 680549 571727 > 3¹⁷¹ [i]