MinT - Best Known (56, 203, s)-Nets in Base 4

Best Known (56, 203, s)-Nets in Base 4

Digital (56, 203, 66)-net over F₄, using

t-expansion [i] based on digital (49, 203, 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, 203, 91)-net over F₄, using

t-expansion [i] based on digital (50, 203, 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, 203, 324)-net over F₄, because

3 times m-reduction [i] would yield digital (56, 200, 324)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁰, 324, F₄, 144) (dual of [324, 124, 145]-code), but
  - residual code [i] would yield OA(4⁵⁶, 179, S₄, 36), but
    - the linear programming bound shows that MÂ â‰¥ 31 511442 125419 258293 449040 901884 044621 696169 017943 480590 506370 111897 600000 / 5839 126320 766662 475522 447173 784673 305301 > 4⁵⁶ [i]

There is no (56, 203, 376)-net in base 4, because

1 times m-reduction [i] would yield (56, 202, 376)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46 581339 391138 998265 592900 110868 548417 154474 243850 606545 830993 690714 607438 633548 899718 427981 118093 005506 589514 249046 688434 > 4²⁰² [i]