Best Known (192−124, 192, s)-Nets in Base 3
(192−124, 192, 48)-Net over F3 — Constructive and digital
Digital (68, 192, 48)-net over F3, using
- t-expansion [i] based on digital (45, 192, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(192−124, 192, 72)-Net over F3 — Digital
Digital (68, 192, 72)-net over F3, using
- t-expansion [i] based on digital (67, 192, 72)-net over F3, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 67 and N(F) ≥ 72, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
(192−124, 192, 258)-Net over F3 — Upper bound on s (digital)
There is no digital (68, 192, 259)-net over F3, because
- 1 times m-reduction [i] would yield digital (68, 191, 259)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3191, 259, F3, 123) (dual of [259, 68, 124]-code), but
- residual code [i] would yield OA(368, 135, S3, 41), but
- the linear programming bound shows that M ≥ 4 012864 604332 264726 237245 813410 565913 537110 032972 476971 418053 506641 356122 698876 582999 914606 332022 203278 572145 154882 258603 361198 432722 111949 572177 753464 008379 174152 471906 949796 598337 565413 664824 822296 942408 625040 904817 451650 047319 073006 184830 147991 735484 840189 024895 041134 841819 175176 574928 936653 856763 351431 / 13772 480989 359896 672406 531732 728032 421482 299092 416728 956572 749201 295898 602651 327734 942411 572072 576928 484223 936120 935396 753748 368299 969660 607387 692434 163811 948112 773848 174076 852121 503697 264927 503103 942619 033909 444632 757394 085121 385194 079177 476674 165653 331143 617081 864855 946660 > 368 [i]
- residual code [i] would yield OA(368, 135, S3, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(3191, 259, F3, 123) (dual of [259, 68, 124]-code), but
(192−124, 192, 290)-Net in Base 3 — Upper bound on s
There is no (68, 192, 291)-net in base 3, because
- 15 times m-reduction [i] would yield (68, 177, 291)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3177, 291, S3, 109), but
- 9 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- the linear programming bound shows that M ≥ 744 646766 850603 526568 601268 797422 932026 535817 992617 489413 629925 481692 835741 945421 210090 969950 381274 252816 504020 353660 158683 457200 925516 493661 440389 831805 338854 508258 811020 990136 433923 762970 235494 388055 889637 880458 633459 997426 817105 857222 614437 226587 / 10512 885684 310613 723066 595609 767913 116130 406095 478720 857818 594175 294259 025889 504412 156455 324830 169263 105843 888569 812652 562285 873819 443701 107860 316189 843301 339995 > 3186 [i]
- 9 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3177, 291, S3, 109), but