Best Known (44, 44+∞, s)-Nets in Base 7
(44, 44+∞, 45)-Net over F7 — Constructive and digital
Digital (44, m, 45)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (44, 44)-sequence over F7, using
- base reduction for sequences [i] based on digital (0, 44)-sequence over F49, using
- s-reduction based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- s-reduction based on digital (0, 49)-sequence over F49, using
- base reduction for sequences [i] based on digital (0, 44)-sequence over F49, using
(44, 44+∞, 105)-Net over F7 — Digital
Digital (44, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (44, 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
(44, 44+∞, 284)-Net in Base 7 — Upper bound on s
There is no (44, m, 285)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (44, 851, 285)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7851, 285, S7, 3, 807), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 32 725914 090844 253821 244252 407174 836676 010383 663702 736801 873951 404042 739639 897470 680987 177656 345545 259133 618594 746690 880640 330971 130726 562081 418135 338881 243710 488817 015563 949957 180910 689341 748461 999213 284149 102205 007637 533988 754935 458022 159874 367669 364357 416984 830899 435150 982893 163239 722146 771447 068960 069378 065081 484419 239884 873078 666601 744819 397005 485535 511218 718087 830152 376044 469917 405996 374688 859125 343720 398219 320274 710248 547984 998555 992108 575384 465451 580401 813918 793025 941163 403540 490558 621625 335847 962363 032699 748828 482666 188167 787301 196044 222999 916856 538443 619786 002641 974519 205262 663839 643970 099305 284990 143758 311292 763837 901983 590484 157102 297277 515813 594454 762630 773567 544984 904421 847258 410919 591385 167927 527231 / 202 > 7851 [i]
- extracting embedded OOA [i] would yield OOA(7851, 285, S7, 3, 807), but