MinT - Best Known (58−39, 58, s)-Nets in Base 4

Best Known (58−39, 58, s)-Nets in Base 4

(58−39, 58, 33)-Net over F₄ — Constructive and digital

Digital (19, 58, 33)-net over F₄, using

t-expansion [i] based on digital (15, 58, 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]

(58−39, 58, 41)-Net over F₄ — Digital

Digital (19, 58, 41)-net over F₄, using

t-expansion [i] based on digital (18, 58, 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]

(58−39, 58, 148)-Net in Base 4 — Upper bound on s

There is no (19, 58, 149)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁸, 149, S₄, 39), but
- the linear programming bound shows that MÂ â‰¥ 687 429911 202260 642762 810713 800269 676548 725551 961062 930193 746529 711584 285763 934617 600000 / 7739 058251 757365 326970 583055 442734 658951 141232 199899 > 4⁵⁸ [i]