MinT - Best Known (251−230, 251, s)-Nets in Base 4

Best Known (251−230, 251, s)-Nets in Base 4

(251−230, 251, 34)-Net over F₄ — Constructive and digital

Digital (21, 251, 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]

(251−230, 251, 44)-Net over F₄ — Digital

Digital (21, 251, 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]

(251−230, 251, 75)-Net in Base 4 — Upper bound on s

There is no (21, 251, 76)-net in base 4, because

27 times m-reduction [i] would yield (21, 224, 76)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²²⁴, 76, S₄, 3, 203), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 14536 774485 912137 810986 476157 760090 687072 827213 746361 205629 803983 612785 762267 958466 523821 014275 271311 215250 432125 323558 670692 032572 293120 / 17 > 4²²⁴ [i]