MinT - Best Known (56, 199, s)-Nets in Base 4

Best Known (56, 199, s)-Nets in Base 4

Digital (56, 199, 66)-net over F₄, using

t-expansion [i] based on digital (49, 199, 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 (56, 199, 91)-net over F₄, using

t-expansion [i] based on digital (50, 199, 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 (56, 199, 345)-net over F₄, because

3 times m-reduction [i] would yield digital (56, 196, 345)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁶, 345, F₄, 140) (dual of [345, 149, 141]-code), but
  - residual code [i] would yield OA(4⁵⁶, 204, S₄, 35), but
    - the linear programming bound shows that MÂ â‰¥ 78295 866271 954113 495370 355913 581218 786159 455204 222712 111589 032673 922357 657600 / 14 695871 573792 476851 963908 736113 155530 258961 > 4⁵⁶ [i]

There is no (56, 199, 379)-net in base 4, because

1 times m-reduction [i] would yield (56, 198, 379)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 185142 893133 861575 750058 796333 311597 129050 342782 744674 762284 038076 822274 561843 564274 421112 341181 827111 721242 225209 877248 > 4¹⁹⁸ [i]