MinT - Best Known (57, 219, s)-Nets in Base 4

Best Known (57, 219, s)-Nets in Base 4

Digital (57, 219, 66)-net over F₄, using

t-expansion [i] based on digital (49, 219, 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, 219, 91)-net over F₄, using

t-expansion [i] based on digital (50, 219, 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, 219, 276)-net over F₄, because

2 times m-reduction [i] would yield digital (57, 217, 276)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁷, 276, F₄, 160) (dual of [276, 59, 161]-code), but
  - residual code [i] would yield OA(4⁵⁷, 115, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 1342 360890 983125 604419 240171 519716 648945 101876 361490 553050 838026 357519 821508 093001 281902 496966 687169 476434 135922 969794 132259 611410 858450 105459 321583 951843 591284 753316 387911 582182 210225 496765 142083 720215 294444 256689 712753 634383 661128 220672 / 62556 178955 518968 237954 851777 165822 720635 469489 559531 576198 945246 246749 285798 445456 187201 128306 946569 457537 344547 604678 005658 660268 156514 242241 855788 497423 435131 135411 095940 105899 895817 042024 977187 310437 > 4⁵⁷ [i]

There is no (57, 219, 375)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 747188 325684 293644 114907 308361 917306 953735 935841 559926 826492 152263 595732 998729 958645 589404 048306 797042 913964 908908 077020 192314 654086 > 4²¹⁹ [i]