Best Known (206−134, 206, s)-Nets in Base 3
(206−134, 206, 48)-Net over F3 — Constructive and digital
Digital (72, 206, 48)-net over F3, using
- t-expansion [i] based on digital (45, 206, 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
(206−134, 206, 84)-Net over F3 — Digital
Digital (72, 206, 84)-net over F3, using
- t-expansion [i] based on digital (71, 206, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(206−134, 206, 261)-Net over F3 — Upper bound on s (digital)
There is no digital (72, 206, 262)-net over F3, because
- 2 times m-reduction [i] would yield digital (72, 204, 262)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3204, 262, F3, 132) (dual of [262, 58, 133]-code), but
- residual code [i] would yield OA(372, 129, S3, 44), but
- the linear programming bound shows that M ≥ 44581 042680 190201 424193 831840 753900 670798 725450 637402 342191 554654 268694 860799 351692 628249 035083 768310 785141 170441 602743 910081 150638 796780 311369 419178 341421 989330 024725 427166 876875 682423 121951 240264 500151 / 1 947482 853721 425586 922187 925058 224422 649757 526790 464738 458184 631778 914633 064948 606561 512864 131931 488340 022430 458898 125986 767827 041137 500285 991795 319155 347570 902673 254730 > 372 [i]
- residual code [i] would yield OA(372, 129, S3, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(3204, 262, F3, 132) (dual of [262, 58, 133]-code), but
(206−134, 206, 294)-Net in Base 3 — Upper bound on s
There is no (72, 206, 295)-net in base 3, because
- 25 times m-reduction [i] would yield (72, 181, 295)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3181, 295, S3, 109), but
- 5 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]
- 5 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3181, 295, S3, 109), but