Best Known (80, 225, s)-Nets in Base 3
(80, 225, 55)-Net over F3 — Constructive and digital
Digital (80, 225, 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
(80, 225, 84)-Net over F3 — Digital
Digital (80, 225, 84)-net over F3, using
- t-expansion [i] based on digital (71, 225, 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
(80, 225, 341)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 225, 342)-net over F3, because
- 1 times m-reduction [i] would yield digital (80, 224, 342)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3224, 342, F3, 144) (dual of [342, 118, 145]-code), but
- residual code [i] would yield OA(380, 197, S3, 48), but
- 6 times truncation [i] would yield OA(374, 191, S3, 42), but
- the linear programming bound shows that M ≥ 7425 373872 326041 246958 413277 162396 664697 868093 270451 580073 051722 485491 109207 030427 537458 984375 / 32911 891840 780579 755648 777011 603243 208306 897748 202179 983781 > 374 [i]
- 6 times truncation [i] would yield OA(374, 191, S3, 42), but
- residual code [i] would yield OA(380, 197, S3, 48), but
- extracting embedded orthogonal array [i] would yield linear OA(3224, 342, F3, 144) (dual of [342, 118, 145]-code), but
(80, 225, 354)-Net in Base 3 — Upper bound on s
There is no (80, 225, 355)-net in base 3, because
- 1 times m-reduction [i] would yield (80, 224, 355)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 79513 238056 313613 317582 965393 915536 490395 520805 635204 229211 644242 072486 292074 897745 884126 352280 949900 396401 > 3224 [i]