Best Known (234, 234+∞, s)-Nets in Base 4
(234, 234+∞, 258)-Net over F4 — Constructive and digital
Digital (234, m, 258)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (234, 257)-sequence over F4, using
- t-expansion [i] based on digital (225, 257)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 225 and N(F) ≥ 258, using
- T8 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 225 and N(F) ≥ 258, using
- t-expansion [i] based on digital (225, 257)-sequence over F4, using
(234, 234+∞, 301)-Net over F4 — Digital
Digital (234, m, 301)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (234, 300)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 234 and N(F) ≥ 301, using
(234, 234+∞, 720)-Net in Base 4 — Upper bound on s
There is no (234, m, 721)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (234, 3599, 721)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(43599, 721, S4, 5, 3365), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1628 728883 078105 336711 024187 777148 189822 371600 420594 860777 683547 642273 667674 171689 063184 924933 198946 470650 589331 243170 545321 928568 442364 536372 865451 751943 477143 587543 840787 309945 503121 428766 207814 904219 594603 889422 974482 356276 307044 359265 457216 511378 570230 042274 058770 710081 229928 515058 166455 293015 674814 567127 132037 903854 687443 534066 319999 782943 969963 236181 269534 333849 882319 491296 174664 699173 688194 325512 449297 854921 928617 079640 317871 640366 394295 721278 653435 747032 435840 904778 110135 297414 569893 889233 659865 346735 499646 226668 550164 683520 385670 916305 376466 851914 381925 339011 797758 876608 418251 899321 454232 178593 958338 618232 381070 090997 686837 299112 602079 124687 846706 414378 362890 302597 277274 917794 647496 657996 590284 542779 085982 595728 552489 172627 574233 402976 787550 646668 845333 103740 093116 115169 213209 574657 201662 284336 542502 364126 661260 822501 069962 801952 749114 214581 423637 364675 771773 644745 501307 594320 410503 310246 245542 045924 380870 280401 698054 230508 993163 593243 109463 740169 337811 114426 109398 667340 189890 396822 533220 697167 033095 197718 221364 025482 818085 118579 200163 933628 255079 427345 205019 777451 253726 565302 446023 874097 048895 186030 899476 758563 055643 143600 110980 688101 548984 608015 160525 984739 041689 507230 438571 189384 427342 970683 850740 057879 870385 243202 079543 950635 633758 699412 200740 796334 769455 568004 817717 725458 761429 489535 109219 845407 194444 146512 883142 097746 664965 077754 529220 000381 990027 479037 341186 757779 674071 726749 129905 970633 221685 923267 047366 536034 256227 327618 696126 056273 230071 727410 652125 301528 044189 287406 467920 713198 740163 564091 806420 133580 257047 023484 123142 649928 300016 568601 189360 934805 306950 724917 193399 432338 934125 161774 950869 432925 728267 538234 729577 215880 147397 526350 750083 642064 375100 320560 182124 625190 060127 104679 894470 848272 254092 526805 136977 263943 194052 737207 134365 629250 008027 710968 860607 274684 156898 943073 127662 194836 953237 956604 012499 400716 486287 837717 630377 712561 979424 986708 013942 612339 212589 482395 732770 272771 661000 791385 386320 888926 983792 954806 176075 737619 101401 659598 208994 290923 183886 141009 771751 482572 365323 113443 581501 727992 725629 283333 209693 732513 432268 883597 364596 570624 267935 784786 388139 164904 836840 897948 001370 541748 409788 174586 490126 336000 / 187 > 43599 [i]
- extracting embedded OOA [i] would yield OOA(43599, 721, S4, 5, 3365), but