MinT - Best Known (19, 19+40, s)-Nets in Base 4

Best Known (19, 19+40, s)-Nets in Base 4

(19, 19+40, 33)-Net over F₄ — Constructive and digital

Digital (19, 59, 33)-net over F₄, using

t-expansion [i] based on digital (15, 59, 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+40, 41)-Net over F₄ — Digital

Digital (19, 59, 41)-net over F₄, using

t-expansion [i] based on digital (18, 59, 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+40, 143)-Net in Base 4 — Upper bound on s

There is no (19, 59, 144)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁹, 144, S₄, 40), but
- the linear programming bound shows that MÂ â‰¥ 267 525938 139566 369814 585811 354664 265574 017946 230036 512794 828836 515572 789957 038790 441318 522730 123561 422710 374400 / 777 477803 949531 233536 373576 188373 355366 920576 587906 617821 062112 199463 182107 > 4⁵⁹ [i]