MinT - Best Known (21, 21+40, s)-Nets in Base 4

Best Known (21, 21+40, s)-Nets in Base 4

(21, 21+40, 34)-Net over F₄ — Constructive and digital

Digital (21, 61, 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, 21+40, 44)-Net over F₄ — Digital

Digital (21, 61, 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, 21+40, 174)-Net in Base 4 — Upper bound on s

There is no (21, 61, 175)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 773082 292841 212711 154498 875327 230866 > 4⁶¹ [i]
extracting embedded orthogonal array [i] would yield OA(4⁶¹, 175, S₄, 40), but
- the linear programming bound shows that MÂ â‰¥ 1 798315 977868 370852 713534 848248 812726 194497 404051 298068 679657 800761 225918 150102 236050 489344 / 313133 764813 349339 900431 732581 184165 925845 010666 409433 > 4⁶¹ [i]