MinT - Best Known (224−155, 224, s)-Nets in Base 4

Best Known (224−155, 224, s)-Nets in Base 4

(224−155, 224, 66)-Net over F₄ — Constructive and digital

Digital (69, 224, 66)-net over F₄, using

t-expansion [i] based on digital (49, 224, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(224−155, 224, 99)-Net over F₄ — Digital

Digital (69, 224, 99)-net over F₄, using

t-expansion [i] based on digital (61, 224, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(224−155, 224, 483)-Net in Base 4 — Upper bound on s

There is no (69, 224, 484)-net in base 4, because

1 times m-reduction [i] would yield (69, 223, 484)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 189 856625 862969 127615 981148 654713 393815 407679 580748 818965 230711 372704 556313 564499 228123 627292 327783 598307 723131 850504 296669 235218 797560 > 4²²³ [i]