Best Known (123−94, 123, s)-Nets in Base 5
(123−94, 123, 51)-Net over F5 — Constructive and digital
Digital (29, 123, 51)-net over F5, using
- t-expansion [i] based on digital (22, 123, 51)-net over F5, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 22 and N(F) ≥ 51, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
(123−94, 123, 56)-Net over F5 — Digital
Digital (29, 123, 56)-net over F5, using
- net from sequence [i] based on digital (29, 55)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 29 and N(F) ≥ 56, using
(123−94, 123, 217)-Net over F5 — Upper bound on s (digital)
There is no digital (29, 123, 218)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(5123, 218, F5, 94) (dual of [218, 95, 95]-code), but
- construction Y1 [i] would yield
- OA(5122, 146, S5, 94), but
- the linear programming bound shows that M ≥ 211 795989 599382 257647 285884 795458 147343 177364 572503 310245 779281 108247 094067 564830 766059 458255 767822 265625 / 9 474927 760905 821721 > 5122 [i]
- OA(595, 218, S5, 72), but
- discarding factors would yield OA(595, 145, S5, 72), but
- the linear programming bound shows that M ≥ 681 594308 011084 642541 521008 533375 736738 792368 662881 901785 086204 561152 741164 839513 298410 490771 406244 903118 931688 368320 465087 890625 / 251 826768 360176 313961 784872 463983 728554 549094 621216 187309 311959 > 595 [i]
- discarding factors would yield OA(595, 145, S5, 72), but
- OA(5122, 146, S5, 94), but
- construction Y1 [i] would yield
(123−94, 123, 276)-Net in Base 5 — Upper bound on s
There is no (29, 123, 277)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 101 602272 954304 723630 806409 035219 158151 467971 289945 473724 680826 015922 071314 676159 284125 > 5123 [i]