MinT - Best Known (224−111, 224, s)-Nets in Base 4

Best Known (224−111, 224, s)-Nets in Base 4

(224−111, 224, 130)-Net over F₄ — Constructive and digital

Digital (113, 224, 130)-net over F₄, using

t-expansion [i] based on digital (105, 224, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(224−111, 224, 165)-Net over F₄ — Digital

Digital (113, 224, 165)-net over F₄, using

t-expansion [i] based on digital (109, 224, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(224−111, 224, 1918)-Net in Base 4 — Upper bound on s

There is no (113, 224, 1919)-net in base 4, because

1 times m-reduction [i] would yield (113, 223, 1919)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 976954 422468 001082 830339 415653 075230 204676 256457 577089 651911 406810 288179 419013 552690 696771 879808 295551 123396 422926 814277 489630 847500 > 4²²³ [i]