MinT - Best Known (223−55, 223, s)-Nets in Base 4

Best Known (223−55, 223, s)-Nets in Base 4

(223−55, 223, 1028)-Net over F₄ — Constructive and digital

Digital (168, 223, 1028)-net over F₄, using

1 times m-reduction [i] based on digital (168, 224, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 56, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(223−55, 223, 2168)-Net over F₄ — Digital

Digital (168, 223, 2168)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−55, 223, 324733)-Net in Base 4 — Upper bound on s

There is no (168, 223, 324734)-net in base 4, because

1 times m-reduction [i] would yield (168, 222, 324734)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 429193 051832 789410 644077 025792 653866 643988 698921 778420 488644 110541 745224 002191 610017 377553 191954 293349 991881 336666 225769 384136 047110 > 4²²² [i]