MinT - Best Known (224−91, 224, s)-Nets in Base 4

Best Known (224−91, 224, s)-Nets in Base 4

(224−91, 224, 137)-Net over F₄ — Constructive and digital

Digital (133, 224, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 60, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 164, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(224−91, 224, 307)-Net over F₄ — Digital

Digital (133, 224, 307)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−91, 224, 5620)-Net in Base 4 — Upper bound on s

There is no (133, 224, 5621)-net in base 4, because

1 times m-reduction [i] would yield (133, 223, 5621)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 876569 477823 581461 691498 091929 941913 565290 342401 453075 719718 027959 271753 224787 656771 174144 833129 551145 438714 452378 973301 867781 218296 > 4²²³ [i]