MinT - Best Known (239−181, 239, s)-Nets in Base 4

Best Known (239−181, 239, s)-Nets in Base 4

(239−181, 239, 66)-Net over F₄ — Constructive and digital

Digital (58, 239, 66)-net over F₄, using

t-expansion [i] based on digital (49, 239, 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]

(239−181, 239, 91)-Net over F₄ — Digital

Digital (58, 239, 91)-net over F₄, using

t-expansion [i] based on digital (50, 239, 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]

(239−181, 239, 240)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 239, 241)-net over F₄, because

5 times m-reduction [i] would yield digital (58, 234, 241)-net over F₄, but
- extracting embedded orthogonal array [i] would yield 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]

(239−181, 239, 376)-Net in Base 4 — Upper bound on s

There is no (58, 239, 377)-net in base 4, because

1 times m-reduction [i] would yield (58, 238, 377)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 203668 064678 803125 371608 450169 796127 288608 288916 215677 892313 565073 276521 748212 006038 269299 770698 111421 162587 224893 061953 623276 569928 617917 952896 > 4²³⁸ [i]