MinT - Best Known (238−179, 238, s)-Nets in Base 4

Best Known (238−179, 238, s)-Nets in Base 4

(238−179, 238, 66)-Net over F₄ — Constructive and digital

Digital (59, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 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]

(238−179, 238, 91)-Net over F₄ — Digital

Digital (59, 238, 91)-net over F₄, using

t-expansion [i] based on digital (50, 238, 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]

(238−179, 238, 246)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 238, 247)-net over F₄, because

3 times m-reduction [i] would yield digital (59, 235, 247)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁵, 247, F₄, 176) (dual of [247, 12, 177]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²³⁴, 241, F₄, 176) (dual of [241, 7, 177]-code), but
      - residual code [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, 247, 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]

(238−179, 238, 384)-Net in Base 4 — Upper bound on s

There is no (59, 238, 385)-net in base 4, because

1 times m-reduction [i] would yield (59, 237, 385)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 58015 449584 639425 945436 076862 402406 590664 608879 996882 595900 275734 515675 214728 282525 481442 469475 432944 960670 340806 161560 356729 745214 769516 806080 > 4²³⁷ [i]