MinT - Best Known (212−157, 212, s)-Nets in Base 4

Best Known (212−157, 212, s)-Nets in Base 4

Digital (55, 212, 66)-net over F₄, using

t-expansion [i] based on digital (49, 212, 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 (55, 212, 91)-net over F₄, using

t-expansion [i] based on digital (50, 212, 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 (55, 212, 261)-net over F₄, because

1 times m-reduction [i] would yield digital (55, 211, 261)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹¹, 261, F₄, 156) (dual of [261, 50, 157]-code), but
  - residual code [i] would yield OA(4⁵⁵, 104, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 7 723473 632829 389285 976355 541453 254660 289881 978452 313500 994274 223551 712140 212225 490113 864437 287145 948134 332311 964905 619217 330530 016843 066600 998037 356544 / 5779 127016 748909 094986 667927 666304 897550 201537 672172 684327 799525 226094 950381 356180 131377 653225 244035 788465 702327 464125 > 4⁵⁵ [i]

There is no (55, 212, 363)-net in base 4, because

1 times m-reduction [i] would yield (55, 211, 363)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 539194 809457 582738 883088 965405 389942 268547 249355 539765 050304 885957 608870 150877 200784 027769 933236 392288 377743 783023 277833 543560 > 4²¹¹ [i]