Best Known (238, 238+∞, s)-Nets in Base 3
(238, 238+∞, 112)-Net over F3 — Constructive and digital
Digital (238, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (238, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(238, 238+∞, 163)-Net over F3 — Digital
Digital (238, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (238, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(238, 238+∞, 491)-Net in Base 3 — Upper bound on s
There is no (238, m, 492)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (238, 2944, 492)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32944, 492, S3, 6, 2706), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 19193 913571 263469 957945 656191 177810 201356 877509 649503 656426 795034 306246 781585 428680 915624 733749 163398 739271 184541 457745 140082 347993 873120 988755 938265 059273 012871 558696 281551 869152 304887 419410 665812 422846 102411 260216 700752 918354 488105 593761 271429 725609 321197 103648 263687 236557 733074 608507 132573 676923 347938 751275 145316 289538 227282 353373 245204 471550 424068 980046 055926 635693 897822 384894 666876 592176 248693 084287 615497 848899 624828 439101 053713 884662 173956 370434 338746 451405 670612 985693 797664 436074 140656 939641 406253 081994 032985 844499 430565 472732 103809 748028 169405 755684 617675 691284 024990 062827 062357 030050 310062 111138 454360 076767 333054 732208 107784 812282 453039 953730 030987 213854 117980 951705 919516 739004 529005 782175 595571 841866 862489 388737 606918 842921 740853 564006 654868 791264 797134 906313 214720 873168 387882 767214 041777 448571 062429 388328 760221 834602 438067 668685 223464 528211 945291 266217 317451 438983 674044 200787 332684 089739 358404 294640 254805 800894 655869 813232 290919 736941 885291 258014 027491 376844 383813 482925 925735 533412 657736 189521 763922 176265 147682 990181 928665 778409 960756 233783 213638 533226 237519 249301 784066 084730 152389 996023 110366 034278 949410 097483 842900 232523 677100 625933 914164 968439 382347 518303 358881 310831 766487 902809 589676 898890 222068 546783 867345 226896 456884 530145 161764 462977 480548 840827 946804 440300 098404 904727 071068 221456 161484 002435 506423 306305 403663 150284 469562 021313 272463 524190 744766 418398 091390 674834 899707 / 2707 > 32944 [i]
- extracting embedded OOA [i] would yield OOA(32944, 492, S3, 6, 2706), but