MinT - Best Known (222−178, 222, s)-Nets in Base 3

Best Known (222−178, 222, s)-Nets in Base 3

(222−178, 222, 42)-Net over F₃ — Constructive and digital

Digital (44, 222, 42)-net over F₃, using

t-expansion [i] based on digital (39, 222, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(222−178, 222, 56)-Net over F₃ — Digital

Digital (44, 222, 56)-net over F₃, using

t-expansion [i] based on digital (40, 222, 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]

(222−178, 222, 109)-Net in Base 3 — Upper bound on s

There is no (44, 222, 110)-net in base 3, because

7 times m-reduction [i] would yield (44, 215, 110)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²¹⁵, 110, S₃, 2, 171), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 205 808602 911063 926879 985732 894336 437044 838287 444272 886280 756726 488231 006482 421084 944904 008715 489399 872978 / 43 > 3²¹⁵ [i]