MinT - Best Known (172, 172+75, s)-Nets in Base 3

Best Known (172, 172+75, s)-Nets in Base 3

(172, 172+75, 246)-Net over F₃ — Constructive and digital

Digital (172, 247, 246)-net over F₃, using

3¹ times duplication [i] based on digital (171, 246, 246)-net over F₃, using
- trace code for nets [i] based on digital (7, 82, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(172, 172+75, 514)-Net over F₃ — Digital

Digital (172, 247, 514)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+75, 10855)-Net in Base 3 — Upper bound on s

There is no (172, 247, 10856)-net in base 3, because

1 times m-reduction [i] would yield (172, 246, 10856)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2360 574868 749359 174332 395099 264263 349999 628704 697172 609498 832436 080955 252528 927650 923177 554539 598291 359048 192094 737489 > 3²⁴⁶ [i]