MinT - Best Known (67−44, 67, s)-Nets in Base 4

Best Known (67−44, 67, s)-Nets in Base 4

(67−44, 67, 34)-Net over F₄ — Constructive and digital

Digital (23, 67, 34)-net over F₄, using

t-expansion [i] based on digital (21, 67, 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]

(67−44, 67, 45)-Net over F₄ — Digital

Digital (23, 67, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(67−44, 67, 184)-Net in Base 4 — Upper bound on s

There is no (23, 67, 185)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁷, 185, S₄, 44), but
- the linear programming bound shows that MÂ â‰¥ 27 922736 614479 963563 498030 039555 280180 497392 114764 937434 686895 163379 799248 329552 391127 539087 989670 936576 / 1212 198916 755108 948361 633672 026124 789528 916769 954260 490443 945723 > 4⁶⁷ [i]