MinT - Best Known (57, 224, s)-Nets in Base 4

Best Known (57, 224, s)-Nets in Base 4

(57, 224, 66)-Net over F₄ — Constructive and digital

Digital (57, 224, 66)-net over F₄, using

t-expansion [i] based on digital (49, 224, 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]

(57, 224, 91)-Net over F₄ — Digital

Digital (57, 224, 91)-net over F₄, using

t-expansion [i] based on digital (50, 224, 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]

(57, 224, 256)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 224, 257)-net over F₄, because

3 times m-reduction [i] would yield digital (57, 221, 257)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²¹, 257, F₄, 164) (dual of [257, 36, 165]-code), but
  - residual code [i] would yield OA(4⁵⁷, 92, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 6 100582 649939 373494 201236 613796 043446 468174 965411 145692 768274 219008 / 286 854938 046996 694993 355683 683465 > 4⁵⁷ [i]

(57, 224, 373)-Net in Base 4 — Upper bound on s

There is no (57, 224, 374)-net in base 4, because

1 times m-reduction [i] would yield (57, 223, 374)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 208 538895 756986 973200 111087 928149 911032 167276 580680 268757 604649 463728 220730 933416 736651 838322 050189 087457 376358 257359 193996 047357 610422 > 4²²³ [i]