MinT - Best Known (56, 56+169, s)-Nets in Base 4

Best Known (56, 56+169, s)-Nets in Base 4

(56, 56+169, 66)-Net over F₄ — Constructive and digital

Digital (56, 225, 66)-net over F₄, using

t-expansion [i] based on digital (49, 225, 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]

(56, 56+169, 91)-Net over F₄ — Digital

Digital (56, 225, 91)-net over F₄, using

t-expansion [i] based on digital (50, 225, 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]

(56, 56+169, 235)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 225, 236)-net over F₄, because

1 times m-reduction [i] would yield digital (56, 224, 236)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁴, 236, F₄, 168) (dual of [236, 12, 169]-code), but
  - residual code [i] would yield linear OA(4⁵⁶, 67, F₄, 42) (dual of [67, 11, 43]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(56, 56+169, 365)-Net in Base 4 — Upper bound on s

There is no (56, 225, 366)-net in base 4, because

1 times m-reduction [i] would yield (56, 224, 366)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 796 257052 715181 823455 940537 797283 441185 843110 935358 811446 148386 073651 536793 416102 585110 865312 484899 203725 762407 060999 980993 171825 201138 > 4²²⁴ [i]