MinT - Best Known (57, 57+159, s)-Nets in Base 4

Best Known (57, 57+159, s)-Nets in Base 4

Digital (57, 216, 66)-net over F₄, using

t-expansion [i] based on digital (49, 216, 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 (57, 216, 91)-net over F₄, using

t-expansion [i] based on digital (50, 216, 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 (57, 216, 291)-net over F₄, because

3 times m-reduction [i] would yield digital (57, 213, 291)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹³, 291, F₄, 156) (dual of [291, 78, 157]-code), but
  - residual code [i] would yield OA(4⁵⁷, 134, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 2118 872152 947773 735205 614860 840104 993723 173522 547123 836257 668222 520389 144284 698869 497944 229365 678080 000000 / 96823 132929 101376 060431 157559 643789 049324 330608 249156 649224 680559 609039 > 4⁵⁷ [i]

There is no (57, 216, 377)-net in base 4, because

1 times m-reduction [i] would yield (57, 215, 377)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3067 873263 161998 151136 665320 046258 579378 529213 930571 145912 758446 451858 117740 092310 425208 733936 067237 521951 283889 380102 810076 708016 > 4²¹⁵ [i]