MinT - Best Known (137−119, 137, s)-Nets in Base 4

Best Known (137−119, 137, s)-Nets in Base 4

(137−119, 137, 33)-Net over F₄ — Constructive and digital

Digital (18, 137, 33)-net over F₄, using

t-expansion [i] based on digital (15, 137, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(137−119, 137, 41)-Net over F₄ — Digital

Digital (18, 137, 41)-net over F₄, using

net from sequence [i] based on digital (18, 40)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
  - Drinfeld modules of rank 1 [i]

(137−119, 137, 68)-Net in Base 4 — Upper bound on s

There is no (18, 137, 69)-net in base 4, because

3 times m-reduction [i] would yield (18, 134, 69)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹³⁴, 69, S₄, 2, 116), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 7588 550360 256754 183279 148073 529370 729071 901715 047420 004889 892225 542594 864082 845696 / 13 > 4¹³⁴ [i]