Best Known (221−141, 221, s)-Nets in Base 3
(221−141, 221, 55)-Net over F3 — Constructive and digital
Digital (80, 221, 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
(221−141, 221, 84)-Net over F3 — Digital
Digital (80, 221, 84)-net over F3, using
- t-expansion [i] based on digital (71, 221, 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
(221−141, 221, 347)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 221, 348)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3221, 348, F3, 141) (dual of [348, 127, 142]-code), but
- residual code [i] would yield OA(380, 206, S3, 47), but
- 5 times truncation [i] would yield OA(375, 201, S3, 42), but
- the linear programming bound shows that M ≥ 70 125936 773107 976834 379244 843784 716640 335910 577502 309916 609029 015855 123273 912868 189563 572000 000000 / 109 901819 499502 271926 662008 620205 982068 132289 165070 372767 942049 > 375 [i]
- 5 times truncation [i] would yield OA(375, 201, S3, 42), but
- residual code [i] would yield OA(380, 206, S3, 47), but
(221−141, 221, 359)-Net in Base 3 — Upper bound on s
There is no (80, 221, 360)-net in base 3, because
- 1 times m-reduction [i] would yield (80, 220, 360)-net in base 3, but
- 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]