MinT - Best Known (59−39, 59, s)-Nets in Base 4

Best Known (59−39, 59, s)-Nets in Base 4

(59−39, 59, 33)-Net over F₄ — Constructive and digital

Digital (20, 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]

(59−39, 59, 41)-Net over F₄ — Digital

Digital (20, 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]

(59−39, 59, 164)-Net in Base 4 — Upper bound on s

There is no (20, 59, 165)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁹, 165, S₄, 39), but
- the linear programming bound shows that MÂ â‰¥ 3078 489369 750206 722287 954878 459506 651865 539185 921556 985227 028939 036190 618883 341680 640000 / 8667 454957 234129 743329 832681 955517 588182 622985 872027 > 4⁵⁹ [i]