MinT - Best Known (243−194, 243, s)-Nets in Base 3

Best Known (243−194, 243, s)-Nets in Base 3

(243−194, 243, 48)-Net over F₃ — Constructive and digital

Digital (49, 243, 48)-net over F₃, using

t-expansion [i] based on digital (45, 243, 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]

(243−194, 243, 64)-Net over F₃ — Digital

Digital (49, 243, 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]

(243−194, 243, 121)-Net in Base 3 — Upper bound on s

There is no (49, 243, 122)-net in base 3, because

4 times m-reduction [i] would yield (49, 239, 122)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²³⁹, 122, S₃, 2, 190), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 319 695355 781856 846744 007258 814822 829555 534932 150683 083646 925804 605966 879343 652613 126077 016325 049504 761567 202494 107299 / 191 > 3²³⁹ [i]