MinT - Best Known (64−42, 64, s)-Nets in Base 4

Best Known (64−42, 64, s)-Nets in Base 4

(64−42, 64, 34)-Net over F₄ — Constructive and digital

Digital (22, 64, 34)-net over F₄, using

t-expansion [i] based on digital (21, 64, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(64−42, 64, 44)-Net over F₄ — Digital

Digital (22, 64, 44)-net over F₄, using

t-expansion [i] based on digital (21, 64, 44)-net over F₄, using
- net from sequence [i] based on digital (21, 43)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
    - a curve from the manYPoints database [i]

(64−42, 64, 179)-Net in Base 4 — Upper bound on s

There is no (22, 64, 180)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁴, 180, S₄, 42), but
- the linear programming bound shows that MÂ â‰¥ 387317 878288 940952 121032 892106 078533 601878 434569 146042 672353 835001 098548 766344 071344 765591 401529 344000 / 1135 320053 976181 914391 551986 355178 156290 618852 310180 915372 701779 > 4⁶⁴ [i]