MinT - Best Known (120−86, 120, s)-Nets in Base 4

Best Known (120−86, 120, s)-Nets in Base 4

(120−86, 120, 56)-Net over F₄ — Constructive and digital

Digital (34, 120, 56)-net over F₄, using

t-expansion [i] based on digital (33, 120, 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]

(120−86, 120, 65)-Net over F₄ — Digital

Digital (34, 120, 65)-net over F₄, using

t-expansion [i] based on digital (33, 120, 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]

(120−86, 120, 145)-Net in Base 4 — Upper bound on s

There is no (34, 120, 146)-net in base 4, because

1 times m-reduction [i] would yield (34, 119, 146)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹¹⁹, 146, S₄, 85), but
  - the linear programming bound shows that MÂ â‰¥ 1 306185 861782 156678 344154 323879 967306 788722 965824 745661 222897 819372 462131 226542 873931 415552 / 2 755682 499112 350745 > 4¹¹⁹ [i]