Best Known (231, ∞, s)-Nets in Base 4
(231, ∞, 258)-Net over F4 — Constructive and digital
Digital (231, m, 258)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (231, 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
(231, ∞, 711)-Net in Base 4 — Upper bound on s
There is no (231, m, 712)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (231, 3554, 712)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(43554, 712, S4, 5, 3323), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 978777 777931 428777 531470 606729 745274 287968 798666 005165 441654 778255 411686 375139 629469 324892 351056 763000 004338 255891 875903 028500 620614 266258 957248 587507 256383 376984 122930 216775 862505 566203 658582 041172 248098 179402 782843 403108 399114 940203 843581 556719 091880 217374 982093 955863 419894 897145 118786 130983 112235 494149 432084 374945 423898 868956 870466 469471 982416 375444 987476 454972 991667 533224 060677 211238 490764 896969 836534 776667 130156 875124 409927 536280 789528 528068 387771 404540 053588 553768 034321 588125 172155 394674 352342 554539 180113 489237 456552 061650 762656 235100 961940 470116 781990 252863 481895 390124 228689 832311 152212 061897 442001 999925 334860 507463 710282 592096 357527 395879 622657 476562 153323 737063 404179 290240 364366 571498 714878 822029 865801 026580 431292 691646 832147 253141 701622 865927 808276 418314 583723 003425 804886 992562 422988 765948 473314 236116 348972 865397 423822 248088 921251 755167 302851 855789 428250 104284 193923 091849 558929 941911 145078 013292 029857 692365 487201 749083 306989 250685 497572 945594 956793 848047 518746 227849 273621 608628 539307 157720 290032 211884 037599 929197 663037 267933 170242 624211 334967 091867 896994 445089 904767 050614 472814 735530 670918 266330 240437 724785 717711 409479 631573 610269 095375 307676 250571 547098 858004 352264 512521 133889 329677 502447 286687 759645 636001 183207 544338 926599 661721 245993 145118 321202 270845 279879 008791 385783 810283 423214 598389 913580 133414 731808 959469 612824 522822 632559 946376 679065 217892 127299 643032 179675 741249 836447 026899 960741 333759 937220 180480 585433 801521 189533 401314 484218 347589 761654 319046 218300 746606 256903 448438 987068 882154 756996 045120 194423 397317 043722 468996 527801 278555 040688 701307 078424 830447 654568 816502 457774 898290 526635 946632 094080 142673 882564 778889 328642 837126 571169 999140 807108 366533 613237 284266 314797 361798 341494 839050 066437 735386 359274 457996 855309 830234 001834 081830 086918 206678 961866 447988 278313 497108 777557 862059 858261 562711 934758 847341 633633 277807 094009 268413 417365 875370 708095 431195 727747 211803 501058 755124 482327 560070 571449 774986 814027 199639 089825 705411 426755 352609 331225 674498 624097 812949 859316 057135 866322 287517 352557 177987 949042 257035 498462 246776 420478 056285 179098 138226 297412 793358 266360 503945 745745 844209 045097 935856 992256 / 277 > 43554 [i]
- extracting embedded OOA [i] would yield OOA(43554, 712, S4, 5, 3323), but