Best Known (242, 242+∞, s)-Nets in Base 3
(242, 242+∞, 115)-Net over F3 — Constructive and digital
Digital (242, m, 115)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (242, 114)-sequence over F3, using
- base reduction for sequences [i] based on digital (64, 114)-sequence over F9, using
- s-reduction based on digital (64, 164)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- T4 from the second tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- s-reduction based on digital (64, 164)-sequence over F9, using
- base reduction for sequences [i] based on digital (64, 114)-sequence over F9, using
(242, 242+∞, 163)-Net over F3 — Digital
Digital (242, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (242, 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
(242, 242+∞, 499)-Net in Base 3 — Upper bound on s
There is no (242, m, 500)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (242, 2992, 500)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32992, 500, S3, 6, 2750), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 506 116958 351899 653232 738645 421826 672824 096885 988029 239232 485726 099684 330951 169147 913454 022370 411382 778270 313072 327970 942766 311025 094413 730092 135045 061622 965255 644786 098737 248272 777719 107044 763289 856937 049109 304242 028005 300392 670044 014071 028467 339089 893987 893728 964343 503313 118063 448512 683125 540258 296541 676701 432556 676239 700428 700291 537983 413435 009577 326134 515972 359634 713189 228081 173833 253998 452111 763420 990006 071228 668833 411674 708739 073167 840452 231814 359653 717230 084519 106993 083179 572990 791335 392024 160386 432069 167729 019851 781592 217948 433551 268720 181362 484097 489631 909307 650419 052880 711299 941410 464335 441075 941363 960762 585122 237062 507536 472567 459463 274720 409636 061251 222808 480290 861636 222477 729847 033194 014392 251603 797483 775690 998870 381317 595087 978352 633909 855792 837912 236470 152663 572861 218491 063876 163235 881494 067332 231738 173516 101440 766349 313085 821633 622105 314291 166016 108350 656845 447993 864158 834430 570433 370409 174780 643834 688535 910227 447764 059175 448945 127317 246839 544763 485983 874473 735668 786529 247862 801180 018846 864464 799965 607580 202205 497031 135668 534480 158187 543733 732827 640355 414558 930400 697815 127611 400874 890131 033675 544594 182655 164947 853404 846209 299511 962188 319830 614235 675014 233292 951209 514341 683918 853007 977781 959099 441360 183407 115049 269633 558817 134078 096930 668300 679178 008707 301490 278349 609036 833768 222603 309993 076826 120083 635046 759434 245296 735622 592255 346781 531003 684291 971406 977283 878025 700402 050124 667459 286398 605917 / 917 > 32992 [i]
- extracting embedded OOA [i] would yield OOA(32992, 500, S3, 6, 2750), but