MinT - Best Known (180−159, 180, s)-Nets in Base 4

Best Known (180−159, 180, s)-Nets in Base 4

(180−159, 180, 34)-Net over F₄ — Constructive and digital

Digital (21, 180, 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]

(180−159, 180, 44)-Net over F₄ — Digital

Digital (21, 180, 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]

(180−159, 180, 78)-Net in Base 4 — Upper bound on s

There is no (21, 180, 79)-net in base 4, because

26 times m-reduction [i] would yield (21, 154, 79)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁵⁴, 79, S₄, 2, 133), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 45890 346474 863302 551455 544468 486471 212167 479972 506211 854330 972043 055501 921068 449316 899576 086528 / 67 > 4¹⁵⁴ [i]