MinT - Best Known (240−181, 240, s)-Nets in Base 4

Best Known (240−181, 240, s)-Nets in Base 4

(240−181, 240, 66)-Net over F₄ — Constructive and digital

Digital (59, 240, 66)-net over F₄, using

t-expansion [i] based on digital (49, 240, 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]

(240−181, 240, 91)-Net over F₄ — Digital

Digital (59, 240, 91)-net over F₄, using

t-expansion [i] based on digital (50, 240, 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]

(240−181, 240, 245)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 240, 246)-net over F₄, because

1 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]

(240−181, 240, 383)-Net in Base 4 — Upper bound on s

There is no (59, 240, 384)-net in base 4, because

1 times m-reduction [i] would yield (59, 239, 384)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 834617 539563 222501 641911 728103 411319 087921 491190 481977 637752 753811 978705 352867 939056 827690 328874 050944 400100 169561 867158 898867 356024 620299 896687 > 4²³⁹ [i]