Best Known (164−140, 164, s)-Nets in Base 8
(164−140, 164, 65)-Net over F8 — Constructive and digital
Digital (24, 164, 65)-net over F8, using
- t-expansion [i] based on digital (14, 164, 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
(164−140, 164, 81)-Net over F8 — Digital
Digital (24, 164, 81)-net over F8, using
- net from sequence [i] based on digital (24, 80)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 24 and N(F) ≥ 81, using
(164−140, 164, 352)-Net over F8 — Upper bound on s (digital)
There is no digital (24, 164, 353)-net over F8, because
- 4 times m-reduction [i] would yield digital (24, 160, 353)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(8160, 353, F8, 136) (dual of [353, 193, 137]-code), but
- residual code [i] would yield OA(824, 216, S8, 17), but
- 1 times truncation [i] would yield OA(823, 215, S8, 16), but
- the linear programming bound shows that M ≥ 9069 643604 547071 244335 560849 505648 640000 / 15 008917 595182 591313 > 823 [i]
- 1 times truncation [i] would yield OA(823, 215, S8, 16), but
- residual code [i] would yield OA(824, 216, S8, 17), but
- extracting embedded orthogonal array [i] would yield linear OA(8160, 353, F8, 136) (dual of [353, 193, 137]-code), but
(164−140, 164, 449)-Net in Base 8 — Upper bound on s
There is no (24, 164, 450)-net in base 8, because
- 20 times m-reduction [i] would yield (24, 144, 450)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 11170 651830 433015 653403 055602 867942 032543 601580 266295 219777 697700 093808 528299 244484 622191 586360 733694 092701 701151 245039 147744 656912 > 8144 [i]