MinT - Best Known (78−55, 78, s)-Nets in Base 4

Best Known (78−55, 78, s)-Nets in Base 4

(78−55, 78, 34)-Net over F₄ — Constructive and digital

Digital (23, 78, 34)-net over F₄, using

t-expansion [i] based on 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]

(78−55, 78, 45)-Net over F₄ — Digital

Digital (23, 78, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(78−55, 78, 125)-Net in Base 4 — Upper bound on s

There is no (23, 78, 126)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁸, 126, S₄, 55), but
- the linear programming bound shows that MÂ â‰¥ 2901 389818 753620 372966 758503 692769 919777 415870 206171 489172 131498 601871 871062 872861 714093 115093 024768 / 29875 532781 103953 314180 525267 680849 659832 645598 366247 > 4⁷⁸ [i]