Best Known (181, ∞, s)-Nets in Base 3
(181, ∞, 104)-Net over F3 — Constructive and digital
Digital (181, m, 104)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (181, 103)-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) ≥ 104 from GarcÃa–Stichtenoth tower as constant field extension [i]
(181, ∞, 163)-Net over F3 — Digital
Digital (181, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (181, 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
(181, ∞, 376)-Net in Base 3 — Upper bound on s
There is no (181, m, 377)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (181, 2255, 377)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32255, 377, S3, 6, 2074), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 43977 366855 820811 121600 882441 893863 013652 902293 070543 683790 097379 104295 876916 443061 009707 662005 524142 185277 608986 276262 663121 356769 364612 154720 140308 237365 067467 307607 649027 658736 841052 882142 189950 699212 272602 261464 758694 482904 294518 303884 907435 955125 899205 601813 343795 086030 340081 830361 812483 035066 100610 140004 603893 136042 841335 188411 306194 467372 403560 389814 366255 715400 456417 555160 426083 188006 286424 311556 464638 666358 498378 534847 853582 682502 805100 596633 757455 201523 277798 735046 946805 294967 502931 964232 266707 148532 645502 914568 716980 682768 698767 384360 962899 003645 020640 605806 949661 279040 287996 720384 717631 432234 033543 020077 711096 142528 353287 651208 958225 904678 520551 298260 701959 876182 031199 766280 785404 108980 987252 285849 297889 024182 503600 280369 680540 071506 702590 154439 953036 597413 745544 799808 520444 253082 920196 655441 543407 055797 930745 398952 812784 579008 840870 835380 362781 717558 496634 914721 013317 794757 776859 033702 367141 758254 864292 481098 943669 523318 641384 317437 204110 964107 385012 490480 928577 615364 542303 094552 084416 645378 850364 489822 888375 525430 892303 823890 957665 218252 872800 269301 / 415 > 32255 [i]
- extracting embedded OOA [i] would yield OOA(32255, 377, S3, 6, 2074), but