MinT - Best Known (78−57, 78, s)-Nets in Base 4

Best Known (78−57, 78, s)-Nets in Base 4

(78−57, 78, 34)-Net over F₄ — Constructive and digital

Digital (21, 78, 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]

(78−57, 78, 44)-Net over F₄ — Digital

Digital (21, 78, 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]

(78−57, 78, 96)-Net in Base 4 — Upper bound on s

There is no (21, 78, 97)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁸, 97, S₄, 57), but
- the linear programming bound shows that MÂ â‰¥ 53737 854835 357207 238510 676971 273446 758615 269988 159876 759552 / 396369 081875 > 4⁷⁸ [i]