Best Known (58, 58+∞, s)-Nets in Base 3
(58, 58+∞, 48)-Net over F3 — Constructive and digital
Digital (58, m, 48)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (58, 47)-sequence over F3, using
- t-expansion [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- t-expansion [i] based on digital (45, 47)-sequence over F3, using
(58, 58+∞, 64)-Net over F3 — Digital
Digital (58, m, 64)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (58, 63)-sequence over F3, using
- t-expansion [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- t-expansion [i] based on digital (49, 63)-sequence over F3, using
(58, 58+∞, 128)-Net in Base 3 — Upper bound on s
There is no (58, m, 129)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (58, 510, 129)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3510, 129, S3, 4, 452), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 367143 646813 489387 003627 305718 372040 052627 849470 463242 453506 214949 146868 181895 203000 340849 165934 557733 411564 256597 136270 597916 672925 325876 797862 769095 844151 308650 452540 820590 172783 184125 740074 424299 824593 696603 255244 038588 520731 776910 176294 287379 / 151 > 3510 [i]
- extracting embedded OOA [i] would yield OOA(3510, 129, S3, 4, 452), but