MinT - Best Known (178−168, 178, s)-Nets in Base 3

Best Known (178−168, 178, s)-Nets in Base 3

(178−168, 178, 19)-Net over F₃ — Constructive and digital

Digital (10, 178, 19)-net over F₃, using

t-expansion [i] based on digital (9, 178, 19)-net over F₃, using
- net from sequence [i] based on digital (9, 18)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
    - an explicitly constructive algebraic function field [i]

(178−168, 178, 20)-Net over F₃ — Digital

Digital (10, 178, 20)-net over F₃, using

net from sequence [i] based on digital (10, 19)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 10 and N(F) ≥ 20, using
  - a curve from the manYPoints database [i]

(178−168, 178, 28)-Net in Base 3 — Upper bound on s

There is no (10, 178, 29)-net in base 3, because

95 times m-reduction [i] would yield (10, 83, 29)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁸³, 29, S₃, 3, 73), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 155642 697373 306257 251835 620347 321612 373853 / 37 > 3⁸³ [i]