MinT - Best Known (239−206, 239, s)-Nets in Base 3

Best Known (239−206, 239, s)-Nets in Base 3

(239−206, 239, 38)-Net over F₃ — Constructive and digital

Digital (33, 239, 38)-net over F₃, using

t-expansion [i] based on digital (32, 239, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(239−206, 239, 46)-Net over F₃ — Digital

Digital (33, 239, 46)-net over F₃, using

net from sequence [i] based on digital (33, 45)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
  - a curve from the manYPoints database [i]

(239−206, 239, 78)-Net in Base 3 — Upper bound on s

There is no (33, 239, 79)-net in base 3, because

7 times m-reduction [i] would yield (33, 232, 79)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²³², 79, S₃, 3, 199), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 31007 849281 193248 357485 012147 590580 799611 263874 059962 942573 618659 042664 718592 901468 492041 322350 342011 763317 917983 / 50 > 3²³² [i]