MinT - Best Known (62, 221, s)-Nets in Base 4

Best Known (62, 221, s)-Nets in Base 4

Digital (62, 221, 66)-net over F₄, using

t-expansion [i] based on digital (49, 221, 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 (62, 221, 99)-net over F₄, using

t-expansion [i] based on digital (61, 221, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

There is no digital (62, 221, 372)-net over F₄, because

3 times m-reduction [i] would yield digital (62, 218, 372)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁸, 372, F₄, 156) (dual of [372, 154, 157]-code), but
  - residual code [i] would yield OA(4⁶², 215, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 12718 083503 539266 529007 190946 224090 701310 717998 449746 183117 383672 455579 566080 / 595 821431 971093 361396 591608 721211 764591 > 4⁶² [i]

There is no (62, 221, 417)-net in base 4, because

1 times m-reduction [i] would yield (62, 220, 417)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 153693 506165 686709 630890 163860 504224 033872 458230 688681 630425 850378 266407 731158 755446 052792 867324 299293 052790 561805 739823 624550 596032 > 4²²⁰ [i]