MinT - Best Known (56, 56+157, s)-Nets in Base 4

Best Known (56, 56+157, s)-Nets in Base 4

Digital (56, 213, 66)-net over F₄, using

t-expansion [i] based on digital (49, 213, 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 (56, 213, 91)-net over F₄, using

t-expansion [i] based on digital (50, 213, 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 (56, 213, 277)-net over F₄, because

1 times m-reduction [i] would yield digital (56, 212, 277)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹², 277, F₄, 156) (dual of [277, 65, 157]-code), but
  - residual code [i] would yield OA(4⁵⁶, 120, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 2066 155678 537090 628147 364565 548181 197682 373267 641122 153961 419152 719537 042174 685359 211352 448103 820851 916170 263907 734704 851244 776014 479360 / 396774 979458 174160 104037 110420 139875 589775 407816 452196 802439 392517 223023 686990 593671 943757 732492 839361 > 4⁵⁶ [i]

There is no (56, 213, 370)-net in base 4, because

1 times m-reduction [i] would yield (56, 212, 370)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 402700 630395 970025 166273 096992 313774 799025 806852 912169 534224 630880 681696 678698 904217 898568 007999 858576 187166 209013 209338 836296 > 4²¹² [i]