MinT - Best Known (215−157, 215, s)-Nets in Base 4

Best Known (215−157, 215, s)-Nets in Base 4

Digital (58, 215, 66)-net over F₄, using

t-expansion [i] based on digital (49, 215, 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, 215, 91)-net over F₄, using

t-expansion [i] based on digital (50, 215, 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, 215, 306)-net over F₄, because

1 times m-reduction [i] would yield digital (58, 214, 306)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁴, 306, F₄, 156) (dual of [306, 92, 157]-code), but
  - residual code [i] would yield OA(4⁵⁸, 149, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 687 429911 202260 642762 810713 800269 676548 725551 961062 930193 746529 711584 285763 934617 600000 / 7739 058251 757365 326970 583055 442734 658951 141232 199899 > 4⁵⁸ [i]

There is no (58, 215, 386)-net in base 4, because

1 times m-reduction [i] would yield (58, 214, 386)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 794 928179 641604 397184 110817 151956 899066 278711 176329 775689 094433 153560 624940 881022 367699 565918 211620 419647 459237 606434 715669 761216 > 4²¹⁴ [i]