Best Known (249, 249+∞, s)-Nets in Base 3
(249, 249+∞, 122)-Net over F3 — Constructive and digital
Digital (249, m, 122)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (249, 121)-sequence over F3, using
- base reduction for sequences [i] based on digital (64, 121)-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, 121)-sequence over F9, using
(249, 249+∞, 163)-Net over F3 — Digital
Digital (249, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (249, 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
(249, 249+∞, 513)-Net in Base 3 — Upper bound on s
There is no (249, m, 514)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (249, 3076, 514)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(33076, 514, S3, 6, 2827), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4 478205 659037 913835 396630 715183 704538 811661 982139 388172 250194 480177 537771 448667 291727 600791 164919 701717 234175 216459 770642 875101 035208 303616 217820 275288 188210 243345 964483 086246 892194 366804 599127 879671 486062 579402 571197 843088 299521 926210 091544 470498 592128 849032 412916 336913 968336 794877 423250 424904 047156 385703 317474 990956 121847 746963 804139 375391 262642 878581 646656 319464 179492 096893 576897 829744 548155 658758 933431 137610 431245 480191 316014 419600 811762 009242 325551 061986 888583 857437 566678 825044 570417 677453 624766 454478 436353 030038 883242 009733 158181 012375 923376 323920 195608 009322 973769 044442 113201 540693 778172 833510 863363 764512 150735 668433 649509 005353 347877 195693 800781 557084 384042 314623 229349 809758 443910 773876 951829 235678 874643 747353 613812 039396 886696 576980 215180 163206 351847 162700 935612 462039 571072 437925 438905 673037 600182 911584 401512 375037 041986 197279 672783 943201 196457 814796 934584 337181 211682 000872 699199 800419 912819 171427 996518 104783 330018 611146 278301 420684 288308 125911 903803 066434 816688 118611 803936 028829 261477 502388 667413 243312 155224 390266 951518 482181 028659 530475 521831 625759 720745 821339 654038 808088 030530 554335 062714 549550 386500 173136 874966 059938 312861 837496 299257 239533 800431 753947 522846 940707 206690 355745 205166 251895 395243 746154 579357 785965 523374 790188 692906 068996 732825 420747 934362 038496 329306 467214 627567 487649 498645 377242 551761 981422 270752 827455 200573 597295 142929 842379 555282 832750 567205 482025 425933 755441 268451 248055 371713 791219 909739 278021 347701 660526 429629 444102 / 707 > 33076 [i]
- extracting embedded OOA [i] would yield OOA(33076, 514, S3, 6, 2827), but