MinT - Best Known (124−89, 124, s)-Nets in Base 4

Best Known (124−89, 124, s)-Nets in Base 4

(124−89, 124, 56)-Net over F₄ — Constructive and digital

Digital (35, 124, 56)-net over F₄, using

t-expansion [i] based on digital (33, 124, 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]

(124−89, 124, 65)-Net over F₄ — Digital

Digital (35, 124, 65)-net over F₄, using

t-expansion [i] based on digital (33, 124, 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]

(124−89, 124, 147)-Net in Base 4 — Upper bound on s

There is no (35, 124, 148)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹²⁴, 148, S₄, 89), but
- the linear programming bound shows that MÂ â‰¥ 8665 917257 296778 952342 423789 447146 428779 676886 749927 658316 397291 649757 237344 216595 862069 968896 / 15 234037 624603 222425 > 4¹²⁴ [i]