MinT - Best Known (224−157, 224, s)-Nets in Base 4

Best Known (224−157, 224, s)-Nets in Base 4

(224−157, 224, 66)-Net over F₄ — Constructive and digital

Digital (67, 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]

(224−157, 224, 99)-Net over F₄ — Digital

Digital (67, 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]

(224−157, 224, 462)-Net in Base 4 — Upper bound on s

There is no (67, 224, 463)-net in base 4, because

1 times m-reduction [i] would yield (67, 223, 463)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204 969013 507594 798383 518747 058174 967873 651941 297464 087076 331569 398015 111045 081765 168667 065965 087308 938523 835829 491846 373912 244398 302195 > 4²²³ [i]