MinT - Best Known (54, 54+205, s)-Nets in Base 4

Best Known (54, 54+205, s)-Nets in Base 4

(54, 54+205, 66)-Net over F₄ — Constructive and digital

Digital (54, 259, 66)-net over F₄, using

t-expansion [i] based on digital (49, 259, 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]

(54, 54+205, 91)-Net over F₄ — Digital

Digital (54, 259, 91)-net over F₄, using

t-expansion [i] based on digital (50, 259, 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]

(54, 54+205, 224)-Net over F₄ — Upper bound on s (digital)

There is no digital (54, 259, 225)-net over F₄, because

41 times m-reduction [i] would yield digital (54, 218, 225)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁸, 225, F₄, 164) (dual of [225, 7, 165]-code), but
  - residual code [i] would yield linear OA(4⁵⁴, 60, F₄, 41) (dual of [60, 6, 42]-code), but
    - 1 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]

(54, 54+205, 230)-Net in Base 4 — Upper bound on s

There is no (54, 259, 231)-net in base 4, because

32 times m-reduction [i] would yield (54, 227, 231)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²²⁷, 231, S₄, 173), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 1 488565 707357 402911 845015 158554 633286 356257 506687 627387 456491 927921 949262 056238 946972 039271 861787 782268 441644 249633 132407 878864 135402 815488 / 29 > 4²²⁷ [i]