MinT - Best Known (55, 55+151, s)-Nets in Base 4

Best Known (55, 55+151, s)-Nets in Base 4

Digital (55, 206, 66)-net over F₄, using

t-expansion [i] based on digital (49, 206, 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]

Digital (55, 206, 91)-net over F₄, using

t-expansion [i] based on digital (50, 206, 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]

There is no digital (55, 206, 293)-net over F₄, because

3 times m-reduction [i] would yield digital (55, 203, 293)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰³, 293, F₄, 148) (dual of [293, 90, 149]-code), but
  - residual code [i] would yield OA(4⁵⁵, 144, S₄, 37), but
    - the linear programming bound shows that MÂ â‰¥ 8531 459492 474233 301926 174796 532559 659169 361619 057384 079100 594036 798926 268886 155264 000000 / 6 295882 403116 171983 140577 568447 723788 871703 289647 780291 > 4⁵⁵ [i]

There is no (55, 206, 365)-net in base 4, because

1 times m-reduction [i] would yield (55, 205, 365)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2661 456828 143105 209569 822296 869383 896532 300126 899630 451457 628210 494076 108275 180491 416680 048800 848034 640426 665773 096566 785316 > 4²⁰⁵ [i]