MinT - Best Known (239−191, 239, s)-Nets in Base 3

Best Known (239−191, 239, s)-Nets in Base 3

(239−191, 239, 48)-Net over F₃ — Constructive and digital

Digital (48, 239, 48)-net over F₃, using

t-expansion [i] based on digital (45, 239, 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]

(239−191, 239, 56)-Net over F₃ — Digital

Digital (48, 239, 56)-net over F₃, using

t-expansion [i] based on digital (40, 239, 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]

(239−191, 239, 118)-Net in Base 3 — Upper bound on s

There is no (48, 239, 119)-net in base 3, because

6 times m-reduction [i] would yield (48, 233, 119)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²³³, 119, S₃, 2, 185), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 53156 313053 474140 041402 877967 298138 513619 309498 388507 901554 774844 073139 517587 831088 843499 409743 443448 737116 430828 / 31 > 3²³³ [i]