Best Known (248−164, 248, s)-Nets in Base 3
(248−164, 248, 59)-Net over F3 — Constructive and digital
Digital (84, 248, 59)-net over F3, using
- net from sequence [i] based on digital (84, 58)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 58)-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, 58)-sequence over F9, using
(248−164, 248, 84)-Net over F3 — Digital
Digital (84, 248, 84)-net over F3, using
- t-expansion [i] based on digital (71, 248, 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
(248−164, 248, 268)-Net over F3 — Upper bound on s (digital)
There is no digital (84, 248, 269)-net over F3, because
- 2 times m-reduction [i] would yield digital (84, 246, 269)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3246, 269, F3, 162) (dual of [269, 23, 163]-code), but
- residual code [i] would yield OA(384, 106, S3, 54), but
- the linear programming bound shows that M ≥ 554636 234644 780595 906506 938993 021399 906097 624808 134107 / 34 652730 233275 > 384 [i]
- residual code [i] would yield OA(384, 106, S3, 54), but
- extracting embedded orthogonal array [i] would yield linear OA(3246, 269, F3, 162) (dual of [269, 23, 163]-code), but
(248−164, 248, 358)-Net in Base 3 — Upper bound on s
There is no (84, 248, 359)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 21911 273496 817928 706238 558548 639408 008598 294103 993761 047142 381861 689119 083591 436102 550129 705112 024427 594538 551658 619565 > 3248 [i]