MinT - Best Known (219−157, 219, s)-Nets in Base 4

Best Known (219−157, 219, s)-Nets in Base 4

Digital (62, 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 (62, 219, 99)-net over F₄, using

t-expansion [i] based on digital (61, 219, 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, 219, 372)-net over F₄, because

1 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, 219, 418)-net in base 4, because

1 times m-reduction [i] would yield (62, 218, 418)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 180395 440401 621828 101668 921541 434978 603377 732905 694061 506376 811955 455367 797258 273152 132411 105632 523521 192255 026604 489515 225674 557280 > 4²¹⁸ [i]