MinT - Best Known (51, 51+197, s)-Nets in Base 3

Best Known (51, 51+197, s)-Nets in Base 3

(51, 51+197, 48)-Net over F₃ — Constructive and digital

Digital (51, 248, 48)-net over F₃, using

t-expansion [i] based on digital (45, 248, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(51, 51+197, 64)-Net over F₃ — Digital

Digital (51, 248, 64)-net over F₃, using

t-expansion [i] based on digital (49, 248, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(51, 51+197, 125)-Net in Base 3 — Upper bound on s

There is no (51, 248, 126)-net in base 3, because

1 times m-reduction [i] would yield (51, 247, 126)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁴⁷, 126, S₃, 2, 196), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 716153 733051 169540 307898 602341 497569 311343 837142 335036 933392 894197 976205 305758 468407 429248 816168 018787 798340 006776 536241 / 197 > 3²⁴⁷ [i]