MinT - Best Known (19, 123, s)-Nets in Base 4

Best Known (19, 123, s)-Nets in Base 4

(19, 123, 33)-Net over F₄ — Constructive and digital

Digital (19, 123, 33)-net over F₄, using

t-expansion [i] based on digital (15, 123, 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, 123, 41)-Net over F₄ — Digital

Digital (19, 123, 41)-net over F₄, using

t-expansion [i] based on digital (18, 123, 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, 123, 84)-Net in Base 4 — Upper bound on s

There is no (19, 123, 85)-net in base 4, because

48 times m-reduction [i] would yield (19, 75, 85)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁷⁵, 85, S₄, 56), but
  - the linear programming bound shows that MÂ â‰¥ 135 462933 010108 064231 004037 932390 482384 359247 577088 / 87685 > 4⁷⁵ [i]