MinT - Best Known (56, 227, s)-Nets in Base 4

Best Known (56, 227, s)-Nets in Base 4

(56, 227, 66)-Net over F₄ — Constructive and digital

Digital (56, 227, 66)-net over F₄, using

t-expansion [i] based on digital (49, 227, 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, 227, 91)-Net over F₄ — Digital

Digital (56, 227, 91)-net over F₄, using

t-expansion [i] based on digital (50, 227, 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, 227, 235)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 227, 236)-net over F₄, because

3 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, 227, 365)-Net in Base 4 — Upper bound on s

There is no (56, 227, 366)-net in base 4, because

1 times m-reduction [i] would yield (56, 226, 366)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14052 790369 023614 569875 504478 605566 477730 076829 691781 994710 640845 663759 208326 223138 854372 142763 534424 432610 527093 244426 325719 893661 375350 > 4²²⁶ [i]