Best Known (166−141, 166, s)-Nets in Base 8
(166−141, 166, 65)-Net over F8 — Constructive and digital
Digital (25, 166, 65)-net over F8, using
- t-expansion [i] based on digital (14, 166, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(166−141, 166, 86)-Net over F8 — Digital
Digital (25, 166, 86)-net over F8, using
- net from sequence [i] based on digital (25, 85)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 25 and N(F) ≥ 86, using
(166−141, 166, 414)-Net over F8 — Upper bound on s (digital)
There is no digital (25, 166, 415)-net over F8, because
- 5 times m-reduction [i] would yield digital (25, 161, 415)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(8161, 415, F8, 136) (dual of [415, 254, 137]-code), but
- residual code [i] would yield OA(825, 278, S8, 17), but
- 1 times truncation [i] would yield OA(824, 277, S8, 16), but
- the linear programming bound shows that M ≥ 78638 403871 838919 782431 803777 157693 440000 / 16 504446 602386 301789 > 824 [i]
- 1 times truncation [i] would yield OA(824, 277, S8, 16), but
- residual code [i] would yield OA(825, 278, S8, 17), but
- extracting embedded orthogonal array [i] would yield linear OA(8161, 415, F8, 136) (dual of [415, 254, 137]-code), but
(166−141, 166, 466)-Net in Base 8 — Upper bound on s
There is no (25, 166, 467)-net in base 8, because
- 23 times m-reduction [i] would yield (25, 143, 467)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1387 994308 584460 028106 920653 117892 514369 247959 049737 779053 772907 081132 280836 298479 079100 816384 514316 382881 595992 537248 121101 435776 > 8143 [i]