MinT - Best Known (22, 22+∞, s)-Nets in Base 4

Best Known (22, 22+∞, s)-Nets in Base 4

(22, 22+∞, 34)-Net over F₄ — Constructive and digital

Digital (22, m, 34)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (22, 33)-sequence over F₄, using
- t-expansion [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]

(22, 22+∞, 44)-Net over F₄ — Digital

Digital (22, m, 44)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (22, 43)-sequence over F₄, using
- t-expansion [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]

(22, 22+∞, 78)-Net in Base 4 — Upper bound on s

There is no (22, m, 79)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (22, 233, 79)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²³³, 79, S₄, 3, 211), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 11241 648221 963106 790253 554477 404590 578562 456690 504962 030071 427039 666560 827048 716527 532840 581100 221331 691271 297373 229415 944301 181950 562062 565376 / 53 > 4²³³ [i]