MinT - Best Known (78−56, 78, s)-Nets in Base 4

Best Known (78−56, 78, s)-Nets in Base 4

(78−56, 78, 34)-Net over F₄ — Constructive and digital

Digital (22, 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]

(78−56, 78, 44)-Net over F₄ — Digital

Digital (22, 78, 44)-net over F₄, using

t-expansion [i] based on 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]

(78−56, 78, 107)-Net in Base 4 — Upper bound on s

There is no (22, 78, 108)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁸, 108, S₄, 56), but
- the linear programming bound shows that MÂ â‰¥ 32084 572525 857164 329603 743742 207132 992618 917067 880819 222047 161524 944896 / 279178 065596 108686 789925 > 4⁷⁸ [i]