MinT - Best Known (77−56, 77, s)-Nets in Base 4

Best Known (77−56, 77, s)-Nets in Base 4

(77−56, 77, 34)-Net over F₄ — Constructive and digital

Digital (21, 77, 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]

(77−56, 77, 44)-Net over F₄ — Digital

Digital (21, 77, 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]

(77−56, 77, 97)-Net in Base 4 — Upper bound on s

There is no (21, 77, 98)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁷, 98, S₄, 56), but
- the linear programming bound shows that MÂ â‰¥ 3 401710 762892 546975 536767 091674 504626 001746 116897 316311 924736 / 145 150982 540625 > 4⁷⁷ [i]