MinT - Best Known (231−184, 231, s)-Nets in Base 3

Best Known (231−184, 231, s)-Nets in Base 3

(231−184, 231, 48)-Net over F₃ — Constructive and digital

Digital (47, 231, 48)-net over F₃, using

t-expansion [i] based on digital (45, 231, 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]

(231−184, 231, 56)-Net over F₃ — Digital

Digital (47, 231, 56)-net over F₃, using

t-expansion [i] based on digital (40, 231, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(231−184, 231, 116)-Net in Base 3 — Upper bound on s

There is no (47, 231, 117)-net in base 3, because

2 times m-reduction [i] would yield (47, 229, 117)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²²⁹, 117, S₃, 2, 182), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1476 564251 485392 778927 857721 313837 180933 869708 288569 663932 077079 002031 653266 328641 356763 872492 873429 131586 567523 / 61 > 3²²⁹ [i]