MinT - Best Known (60, 60+183, s)-Nets in Base 4

Best Known (60, 60+183, s)-Nets in Base 4

(60, 60+183, 66)-Net over F₄ — Constructive and digital

Digital (60, 243, 66)-net over F₄, using

t-expansion [i] based on digital (49, 243, 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]

(60, 60+183, 91)-Net over F₄ — Digital

Digital (60, 243, 91)-net over F₄, using

t-expansion [i] based on digital (50, 243, 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]

(60, 60+183, 251)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 243, 252)-net over F₄, because

3 times m-reduction [i] would yield digital (60, 240, 252)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁰, 252, F₄, 180) (dual of [252, 12, 181]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²³⁹, 246, F₄, 180) (dual of [246, 7, 181]-code), but
      - residual code [i] would yield linear OA(4⁵⁹, 65, F₄, 45) (dual of [65, 6, 46]-code), but
        1 times truncation [i] would yield linear OA(4⁵⁸, 64, F₄, 44) (dual of [64, 6, 45]-code), but
        residual code [i] would yield linear OA(4¹⁴, 19, F₄, 11) (dual of [19, 5, 12]-code), but
        1 times truncation [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]
    2. OA(4¹², 252, 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]

(60, 60+183, 390)-Net in Base 4 — Upper bound on s

There is no (60, 243, 391)-net in base 4, because

1 times m-reduction [i] would yield (60, 242, 391)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 58 957505 491035 256226 051412 155481 093034 170587 274240 844704 960696 121335 389131 949292 229954 708491 826409 489766 884315 938202 038733 303211 726198 617157 764992 > 4²⁴² [i]