MinT - Best Known (172, 172+77, s)-Nets in Base 3

Best Known (172, 172+77, s)-Nets in Base 3

(172, 172+77, 228)-Net over F₃ — Constructive and digital

Digital (172, 249, 228)-net over F₃, using

trace code for nets [i] based on digital (6, 83, 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, 172+77, 488)-Net over F₃ — Digital

Digital (172, 249, 488)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+77, 9726)-Net in Base 3 — Upper bound on s

There is no (172, 249, 9727)-net in base 3, because

1 times m-reduction [i] would yield (172, 248, 9727)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21243 119434 185190 741465 893288 086078 933460 611468 422860 045476 945216 070531 456137 304589 147493 120863 293197 472068 887109 950389 > 3²⁴⁸ [i]