Best Known (109−104, 109, s)-Nets in Base 25
(109−104, 109, 66)-Net over F25 — Constructive and digital
Digital (5, 109, 66)-net over F25, using
- t-expansion [i] based on digital (4, 109, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
(109−104, 109, 282)-Net over F25 — Upper bound on s (digital)
There is no digital (5, 109, 283)-net over F25, because
- 4 times m-reduction [i] would yield digital (5, 105, 283)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(25105, 283, F25, 100) (dual of [283, 178, 101]-code), but
- residual code [i] would yield OA(255, 182, S25, 4), but
- the linear programming bound shows that M ≥ 1 005484 453125 / 102709 > 255 [i]
- residual code [i] would yield OA(255, 182, S25, 4), but
- extracting embedded orthogonal array [i] would yield linear OA(25105, 283, F25, 100) (dual of [283, 178, 101]-code), but
(109−104, 109, 299)-Net in Base 25 — Upper bound on s
There is no (5, 109, 300)-net in base 25, because
- 4 times m-reduction [i] would yield (5, 105, 300)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25105, 300, S25, 100), but
- the linear programming bound shows that M ≥ 72745 196255 970074 413602 301692 353543 516029 770741 676280 992841 899000 891697 307963 931938 352854 327873 222317 708949 573911 777717 528973 198671 057961 387149 073653 591679 726460 039745 461472 193710 505962 371826 171875 / 116 817799 986983 288419 764695 938190 235541 426029 458947 > 25105 [i]
- extracting embedded orthogonal array [i] would yield OA(25105, 300, S25, 100), but