Best Known (100, ∞, s)-Nets in Base 7
(100, ∞, 81)-Net over F7 — Constructive and digital
Digital (100, m, 81)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (100, 80)-sequence over F7, using
- Niederreiter–Xing sequence construction III based on function field F2/F7 with g(F2) = 21, N(F2) ≥ 2, and N(F2) + N2(F2) ≥ 81 from GarcÃa–Stichtenoth tower as constant field extension [i]
(100, ∞, 105)-Net over F7 — Digital
Digital (100, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (100, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(100, ∞, 624)-Net in Base 7 — Upper bound on s
There is no (100, m, 625)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (100, 1871, 625)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71871, 625, S7, 3, 1771), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 781200 704769 214975 597862 558625 619235 620914 509843 248621 950591 902595 184117 098275 776270 587785 744998 575391 194155 397522 295718 241286 053410 978861 125539 979664 959495 302089 841047 828149 497894 754275 826521 634241 010508 311789 802914 964906 350493 325437 921292 257113 372919 134145 243776 130568 471376 651639 742376 924118 098915 171912 932354 412450 053770 181846 157077 939512 483012 498700 474705 288656 406116 230098 977350 435573 540198 809753 784308 423965 074897 202289 744535 375343 931318 426494 660286 093612 358600 778239 920900 320202 237845 861729 555265 796248 527844 002759 199265 703441 146249 214833 159428 116946 879430 583854 338597 221198 566130 955085 517329 627502 630180 792867 541382 631605 695541 202846 642047 401217 556027 866223 842591 994651 628754 109133 974344 850704 560547 290075 220811 933177 776812 493079 546092 420494 910602 607925 205983 003122 054897 674844 758440 721246 177281 606893 201734 994690 292544 297644 365082 904175 355636 806400 359019 639926 418167 648273 754577 584153 289462 710257 379995 764497 493458 183104 695853 341725 352668 123755 154844 415121 780791 391776 523521 217134 578074 030976 526643 355658 347344 890529 867996 407204 254571 622737 289764 361190 689748 177355 709861 651276 658194 663497 907485 102590 652452 843354 335015 015451 492378 657900 799563 571540 353103 857972 658235 796486 897335 697583 622670 982018 722014 601310 224269 102695 863276 677883 349222 847732 802086 091083 433426 386329 975216 573679 201291 056481 862986 849223 619955 251872 056902 773622 205070 797659 818058 786573 575873 776845 872202 157636 471556 590683 535099 692613 975171 566397 654908 172045 411452 195538 432732 563253 765065 258086 199710 936188 795589 105227 486201 521921 172130 246898 627940 739582 298651 134167 683190 690729 014365 383455 450668 002353 836874 / 443 > 71871 [i]
- extracting embedded OOA [i] would yield OOA(71871, 625, S7, 3, 1771), but