MinT - Best Known (59, 59+186, s)-Nets in Base 4

Best Known (59, 59+186, s)-Nets in Base 4

(59, 59+186, 66)-Net over F₄ — Constructive and digital

Digital (59, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(59, 59+186, 91)-Net over F₄ — Digital

Digital (59, 245, 91)-net over F₄, using

t-expansion [i] based on digital (50, 245, 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]

(59, 59+186, 245)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 245, 246)-net over F₄, because

6 times m-reduction [i] would yield digital (59, 239, 246)-net over F₄, but
- extracting embedded orthogonal array [i] would yield 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]

(59, 59+186, 382)-Net in Base 4 — Upper bound on s

There is no (59, 245, 383)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3487 750310 976273 700130 833001 312978 680605 772459 330926 691202 177801 059284 488991 801011 018815 232073 547710 533057 281388 810713 665465 986798 579155 440209 545270 > 4²⁴⁵ [i]