MinT - Best Known (45, 45+189, s)-Nets in Base 3

Best Known (45, 45+189, s)-Nets in Base 3

(45, 45+189, 48)-Net over F₃ — Constructive and digital

Digital (45, 234, 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]

(45, 45+189, 56)-Net over F₃ — Digital

Digital (45, 234, 56)-net over F₃, using

t-expansion [i] based on digital (40, 234, 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]

(45, 45+189, 111)-Net in Base 3 — Upper bound on s

There is no (45, 234, 112)-net in base 3, because

15 times m-reduction [i] would yield (45, 219, 112)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²¹⁹, 112, S₃, 2, 174), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 8335 248417 898089 038639 422182 220625 700315 950641 493051 894370 647422 773355 762538 053940 268612 352977 320694 855609 / 25 > 3²¹⁹ [i]