Best Known (237, 237+∞, s)-Nets in Base 3
(237, 237+∞, 112)-Net over F3 — Constructive and digital
Digital (237, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (237, 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
(237, 237+∞, 163)-Net over F3 — Digital
Digital (237, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (237, 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
(237, 237+∞, 489)-Net in Base 3 — Upper bound on s
There is no (237, m, 490)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (237, 2932, 490)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32932, 490, S3, 6, 2695), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 9047 877620 825627 429756 753815 409697 405137 366133 156820 173118 836554 368162 648010 989794 370636 535189 839592 646007 321753 015053 033296 732222 904447 343938 222179 072896 446782 330403 306283 031597 281490 170497 105614 688169 726152 783197 223228 035230 627907 579538 389722 780477 157006 685485 066773 023940 404346 970291 804502 541993 788031 320524 682381 185522 318882 205502 115632 464219 723396 240602 853468 106396 754467 786746 652843 659937 803822 763171 085239 266890 819147 007136 292894 708575 609589 124431 358215 978064 398564 698009 456781 314113 847569 201880 517640 157111 636473 949786 935201 224044 088453 781063 428293 347330 819354 254565 127276 525483 450475 484773 852345 643668 591378 429344 459813 114925 401841 884951 462732 599413 820587 065186 987526 137747 434017 761457 557970 933238 798627 414311 680879 677873 399420 146775 711191 705473 731028 192226 907103 008900 329408 400400 293481 584318 795345 733273 576876 554456 224038 215230 984005 730577 027369 926024 542055 934434 007340 961700 294033 361434 465340 811195 156468 736830 464540 374288 555812 867603 486744 418120 718955 767212 976262 116630 373653 180590 595015 808255 550210 421610 458574 881186 968615 666246 567283 838954 028402 960046 749608 947397 006060 471343 721348 126069 610578 608444 015368 808979 244881 622141 059989 656449 498452 690223 251534 136752 931027 757344 082468 877785 454258 042422 205432 466708 038109 480246 369135 961240 828869 925359 991574 998424 340054 138987 008042 418566 593625 260789 835896 329712 203043 514977 699338 051023 194448 359478 412295 837490 234978 205753 899704 210307 755106 816849 / 674 > 32932 [i]
- extracting embedded OOA [i] would yield OOA(32932, 490, S3, 6, 2695), but