MinT - Best Known (220−161, 220, s)-Nets in Base 4

Best Known (220−161, 220, s)-Nets in Base 4

Digital (59, 220, 66)-net over F₄, using

t-expansion [i] based on digital (49, 220, 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 (59, 220, 91)-net over F₄, using

t-expansion [i] based on digital (50, 220, 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 (59, 220, 305)-net over F₄, because

1 times m-reduction [i] would yield digital (59, 219, 305)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁹, 305, F₄, 160) (dual of [305, 86, 161]-code), but
  - residual code [i] would yield OA(4⁵⁹, 144, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 267 525938 139566 369814 585811 354664 265574 017946 230036 512794 828836 515572 789957 038790 441318 522730 123561 422710 374400 / 777 477803 949531 233536 373576 188373 355366 920576 587906 617821 062112 199463 182107 > 4⁵⁹ [i]

There is no (59, 220, 391)-net in base 4, because

1 times m-reduction [i] would yield (59, 219, 391)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 724110 341184 134368 749416 435334 416425 832968 812673 031711 549979 282020 384542 143046 067417 399805 652875 825429 267614 704522 124825 599142 289112 > 4²¹⁹ [i]