MinT - Best Known (55, 55+168, s)-Nets in Base 4

Best Known (55, 55+168, s)-Nets in Base 4

Digital (55, 223, 66)-net over F₄, using

t-expansion [i] based on digital (49, 223, 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]

Digital (55, 223, 91)-net over F₄, using

t-expansion [i] based on digital (50, 223, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

There is no digital (55, 223, 230)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²³, 230, F₄, 168) (dual of [230, 7, 169]-code), but
- residual code [i] would yield linear OA(4⁵⁵, 61, F₄, 42) (dual of [61, 6, 43]-code), but
  - 2 times truncation [i] would yield linear OA(4⁵³, 59, F₄, 40) (dual of [59, 6, 41]-code), but
    - residual code [i] would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
      - â€œLizâ€ bound on codes from Brouwerâ€™s database [i]

There is no (55, 223, 359)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 183229 338133 600045 268665 491554 252715 458359 124808 279064 673617 377101 935738 252949 867914 555429 538425 098679 711781 029434 361960 059463 860078 > 4²²³ [i]