MinT - Best Known (224−161, 224, s)-Nets in Base 4

Best Known (224−161, 224, s)-Nets in Base 4

Digital (63, 224, 66)-net over F₄, using

t-expansion [i] based on digital (49, 224, 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 (63, 224, 99)-net over F₄, using

t-expansion [i] based on digital (61, 224, 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 (63, 224, 369)-net over F₄, because

1 times m-reduction [i] would yield digital (63, 223, 369)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²³, 369, F₄, 160) (dual of [369, 146, 161]-code), but
  - residual code [i] would yield OA(4⁶³, 208, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 5 853210 937914 115965 830373 633983 054380 602394 595992 687313 196476 039443 785703 042842 624000 / 66248 338389 800632 875754 408934 650630 828021 253383 > 4⁶³ [i]

There is no (63, 224, 424)-net in base 4, because

1 times m-reduction [i] would yield (63, 223, 424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206 349161 011347 351464 571179 956719 500385 019927 530646 350640 847995 723864 518363 157374 235447 157332 377385 186641 613910 797978 948127 302125 600268 > 4²²³ [i]