MinT - Best Known (75−53, 75, s)-Nets in Base 4

Best Known (75−53, 75, s)-Nets in Base 4

(75−53, 75, 34)-Net over F₄ — Constructive and digital

Digital (22, 75, 34)-net over F₄, using

t-expansion [i] based on digital (21, 75, 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]

(75−53, 75, 44)-Net over F₄ — Digital

Digital (22, 75, 44)-net over F₄, using

t-expansion [i] based on digital (21, 75, 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]

(75−53, 75, 121)-Net in Base 4 — Upper bound on s

There is no (22, 75, 122)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁵, 122, S₄, 53), but
- the linear programming bound shows that MÂ â‰¥ 1 347352 177052 742706 523689 292153 536880 634781 153560 243699 700721 949306 115649 792867 256339 571687 817216 / 815 584986 896999 317274 326882 803224 953269 716215 945785 > 4⁷⁵ [i]