MinT - Best Known (55, 55+163, s)-Nets in Base 4

Best Known (55, 55+163, s)-Nets in Base 4

(55, 55+163, 66)-Net over F₄ — Constructive and digital

Digital (55, 218, 66)-net over F₄, using

t-expansion [i] based on digital (49, 218, 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]

(55, 55+163, 91)-Net over F₄ — Digital

Digital (55, 218, 91)-net over F₄, using

t-expansion [i] based on digital (50, 218, 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]

(55, 55+163, 237)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 218, 238)-net over F₄, because

3 times m-reduction [i] would yield digital (55, 215, 238)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁵, 238, F₄, 160) (dual of [238, 23, 161]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²¹⁴, 226, F₄, 160) (dual of [226, 12, 161]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4²¹³, 220, F₄, 160) (dual of [220, 7, 161]-code), but
        residual code [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]
        OA(4¹², 226, S₄, 6), but
        discarding factors would yield OA(4¹², 156, S₄, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 16 866019 > 4¹² [i]
    2. OA(4²³, 238, S₄, 12), but
      - discarding factors would yield OA(4²³, 205, S₄, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 70 503355 038244 > 4²³ [i]

(55, 55+163, 360)-Net in Base 4 — Upper bound on s

There is no (55, 218, 361)-net in base 4, because

1 times m-reduction [i] would yield (55, 217, 361)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52770 919438 455661 239041 764861 341895 189521 386508 961359 114036 225388 101124 411544 430320 951516 124017 971600 376184 907543 502032 862543 258920 > 4²¹⁷ [i]