Best Known (48, 48+∞, s)-Nets in Base 4
(48, 48+∞, 56)-Net over F4 — Constructive and digital
Digital (48, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (48, 55)-sequence over F4, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
(48, 48+∞, 81)-Net over F4 — Digital
Digital (48, m, 81)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (48, 80)-sequence over F4, using
- t-expansion [i] based on digital (46, 80)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 46 and N(F) ≥ 81, using
- t-expansion [i] based on digital (46, 80)-sequence over F4, using
(48, 48+∞, 158)-Net in Base 4 — Upper bound on s
There is no (48, m, 159)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (48, 631, 159)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4631, 159, S4, 4, 583), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 8059 731428 210727 743647 856851 405751 664500 801987 438740 064543 837765 243272 537665 146409 974993 158995 291936 962171 888857 723406 149486 022752 417233 949098 209537 324890 574926 751290 916487 618224 930648 732381 670936 625527 844165 922836 203828 662690 510597 203780 292667 262417 685191 435692 465393 803781 764071 410704 602217 257972 169940 090007 206372 213461 946181 420914 508020 324181 535059 190543 902759 888855 906706 784256 / 73 > 4631 [i]
- extracting embedded OOA [i] would yield OOA(4631, 159, S4, 4, 583), but