MinT - Best Known (19, 19+55, s)-Nets in Base 4

Best Known (19, 19+55, s)-Nets in Base 4

(19, 19+55, 33)-Net over F₄ — Constructive and digital

Digital (19, 74, 33)-net over F₄, using

t-expansion [i] based on digital (15, 74, 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]

(19, 19+55, 41)-Net over F₄ — Digital

Digital (19, 74, 41)-net over F₄, using

t-expansion [i] based on digital (18, 74, 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]

(19, 19+55, 85)-Net in Base 4 — Upper bound on s

There is no (19, 74, 86)-net in base 4, because

1 times m-reduction [i] would yield (19, 73, 86)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁷³, 86, S₄, 54), but
  - the linear programming bound shows that MÂ â‰¥ 6 529660 477726 074785 377468 003488 991368 143284 017194 598400 / 63993 007639 > 4⁷³ [i]