Best Known (38, 38+∞, s)-Nets in Base 7
(38, 38+∞, 39)-Net over F7 — Constructive and digital
Digital (38, m, 39)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (38, 38)-sequence over F7, using
- base reduction for sequences [i] based on digital (0, 38)-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, 38)-sequence over F49, using
(38, 38+∞, 96)-Net over F7 — Digital
Digital (38, m, 96)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (38, 95)-sequence over F7, using
- t-expansion [i] based on digital (33, 95)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 33 and N(F) ≥ 96, using
- t-expansion [i] based on digital (33, 95)-sequence over F7, using
(38, 38+∞, 248)-Net in Base 7 — Upper bound on s
There is no (38, m, 249)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (38, 743, 249)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7743, 249, S7, 3, 705), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3 680061 929839 440849 889523 048368 393040 207429 569304 182019 066149 717598 130950 146264 393733 636617 482050 573581 247841 306917 189837 848689 113348 340034 608511 986645 950319 486390 234892 759423 532314 292883 500644 977188 866505 483287 458316 186565 943539 334309 637149 437885 254326 059917 224233 604855 718357 364636 912512 160660 047179 134685 138944 985178 668547 851294 267255 193636 321543 046054 972590 666291 768138 727711 853881 800017 343510 318626 453729 727064 420023 532671 067786 487512 423853 631746 245354 039171 397989 498312 246573 585675 432954 185924 949998 883927 387922 091331 309771 124524 501641 672643 757101 074989 636200 088090 326440 475445 854531 308393 772824 319507 620746 085539 215981 016065 / 353 > 7743 [i]
- extracting embedded OOA [i] would yield OOA(7743, 249, S7, 3, 705), but