Best Known (220−140, 220, s)-Nets in Base 3
(220−140, 220, 55)-Net over F3 — Constructive and digital
Digital (80, 220, 55)-net over F3, using
- net from sequence [i] based on digital (80, 54)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 54)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 54)-sequence over F9, using
(220−140, 220, 84)-Net over F3 — Digital
Digital (80, 220, 84)-net over F3, using
- t-expansion [i] based on digital (71, 220, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(220−140, 220, 355)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 220, 356)-net over F3, because
- 2 times m-reduction [i] would yield digital (80, 218, 356)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3218, 356, F3, 138) (dual of [356, 138, 139]-code), but
- residual code [i] would yield OA(380, 217, S3, 46), but
- 4 times truncation [i] would yield OA(376, 213, S3, 42), but
- the linear programming bound shows that M ≥ 24139 354386 854363 147378 476124 618112 504428 827791 667688 382538 767752 602863 057296 834401 801307 804700 / 12846 980782 406505 841743 696059 469546 157683 736221 588601 551817 > 376 [i]
- 4 times truncation [i] would yield OA(376, 213, S3, 42), but
- residual code [i] would yield OA(380, 217, S3, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(3218, 356, F3, 138) (dual of [356, 138, 139]-code), but
(220−140, 220, 359)-Net in Base 3 — Upper bound on s
There is no (80, 220, 360)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 981 197023 785887 074730 633036 055290 411661 908442 092778 716187 205813 798732 853089 422401 563912 435229 127176 218737 > 3220 [i]