MinT - Best Known (42, 42+180, s)-Nets in Base 3

Best Known (42, 42+180, s)-Nets in Base 3

(42, 42+180, 42)-Net over F₃ — Constructive and digital

Digital (42, 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]

(42, 42+180, 56)-Net over F₃ — Digital

Digital (42, 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]

(42, 42+180, 105)-Net in Base 3 — Upper bound on s

There is no (42, 222, 106)-net in base 3, because

15 times m-reduction [i] would yield (42, 207, 106)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁰⁷, 106, S₃, 2, 165), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 78421 202145 657646 273428 491424 453755 923197 030728 651457 964013 384578 658362 476154 963018 177110 469246 075245 / 83 > 3²⁰⁷ [i]