MinT - Best Known (82−61, 82, s)-Nets in Base 4

Best Known (82−61, 82, s)-Nets in Base 4

(82−61, 82, 34)-Net over F₄ — Constructive and digital

Digital (21, 82, 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]

(82−61, 82, 44)-Net over F₄ — Digital

Digital (21, 82, 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]

(82−61, 82, 93)-Net in Base 4 — Upper bound on s

There is no (21, 82, 94)-net in base 4, because

1 times m-reduction [i] would yield (21, 81, 94)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁸¹, 94, S₄, 60), but
  - the linear programming bound shows that MÂ â‰¥ 69275 874266 393268 119010 487057 608395 667750 747969 739525 455872 / 8935 180875 > 4⁸¹ [i]