MinT - Best Known (21, 64, s)-Nets in Base 4

Best Known (21, 64, s)-Nets in Base 4

(21, 64, 34)-Net over F₄ — Constructive and digital

Digital (21, 64, 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]

(21, 64, 44)-Net over F₄ — Digital

Digital (21, 64, 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]

(21, 64, 158)-Net in Base 4 — Upper bound on s

There is no (21, 64, 159)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁴, 159, S₄, 43), but
- the linear programming bound shows that MÂ â‰¥ 45270 819771 373261 134814 974786 535179 781633 219686 501979 035125 061347 877742 643588 123815 870907 832622 000802 701749 559691 837440 / 132 156573 038196 016753 662455 730323 092222 536469 928086 508825 109263 446128 015973 837189 > 4⁶⁴ [i]