Best Known (219−162, 219, s)-Nets in Base 4
(219−162, 219, 66)-Net over F4 — Constructive and digital
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
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(219−162, 219, 91)-Net over F4 — Digital
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
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(219−162, 219, 275)-Net over F4 — Upper bound on s (digital)
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 M ≥ 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]
- residual code [i] would yield OA(457, 115, S4, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(4217, 276, F4, 160) (dual of [276, 59, 161]-code), but
(219−162, 219, 374)-Net in Base 4 — Upper bound on s
There is no (57, 219, 375)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 747188 325684 293644 114907 308361 917306 953735 935841 559926 826492 152263 595732 998729 958645 589404 048306 797042 913964 908908 077020 192314 654086 > 4219 [i]