MinT - Best Known (125−91, 125, s)-Nets in Base 4

Best Known (125−91, 125, s)-Nets in Base 4

(125−91, 125, 56)-Net over F₄ — Constructive and digital

Digital (34, 125, 56)-net over F₄, using

t-expansion [i] based on digital (33, 125, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(125−91, 125, 65)-Net over F₄ — Digital

Digital (34, 125, 65)-net over F₄, using

t-expansion [i] based on digital (33, 125, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(125−91, 125, 141)-Net in Base 4 — Upper bound on s

There is no (34, 125, 142)-net in base 4, because

1 times m-reduction [i] would yield (34, 124, 142)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²⁴, 142, S₄, 90), but
  - the linear programming bound shows that MÂ â‰¥ 544 319840 217452 994883 134694 586320 126404 104288 609422 175226 372537 873340 931624 066946 695168 / 1 053375 235055 > 4¹²⁴ [i]