MinT - Best Known (58, 219, s)-Nets in Base 4

Best Known (58, 219, s)-Nets in Base 4

Digital (58, 219, 66)-net over F₄, using

t-expansion [i] based on digital (49, 219, 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 (58, 219, 91)-net over F₄, using

t-expansion [i] based on digital (50, 219, 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 (58, 219, 291)-net over F₄, because

1 times m-reduction [i] would yield digital (58, 218, 291)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁸, 291, F₄, 160) (dual of [291, 73, 161]-code), but
  - residual code [i] would yield OA(4⁵⁸, 130, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 209544 936299 479389 121518 912044 089753 196453 227215 118049 279760 432327 769939 564811 820879 196123 613172 070974 880613 527842 029138 321043 777447 471353 954304 / 2 470482 844369 173135 750828 733014 583913 640615 187243 459992 006489 938479 554355 225141 791107 956885 358375 430667 629325 > 4⁵⁸ [i]

There is no (58, 219, 384)-net in base 4, because

1 times m-reduction [i] would yield (58, 218, 384)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206877 581283 380246 267676 141000 931399 331344 958185 073643 496531 594116 596598 106498 860010 075500 395623 667868 648890 873420 236097 273118 693823 > 4²¹⁸ [i]