MinT - Best Known (220−162, 220, s)-Nets in Base 4

Best Known (220−162, 220, s)-Nets in Base 4

Digital (58, 220, 66)-net over F₄, using

t-expansion [i] based on digital (49, 220, 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, 220, 91)-net over F₄, using

t-expansion [i] based on digital (50, 220, 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, 220, 291)-net over F₄, because

2 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, 220, 383)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 268925 706105 662376 292529 010043 127259 945410 108481 600357 189131 232876 368058 799483 032980 625668 940464 159002 624484 477036 709513 654564 528370 > 4²²⁰ [i]