Best Known (84, 84+160, s)-Nets in Base 3
(84, 84+160, 59)-Net over F3 — Constructive and digital
Digital (84, 244, 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
(84, 84+160, 84)-Net over F3 — Digital
Digital (84, 244, 84)-net over F3, using
- t-expansion [i] based on digital (71, 244, 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
(84, 84+160, 272)-Net over F3 — Upper bound on s (digital)
There is no digital (84, 244, 273)-net over F3, because
- 1 times m-reduction [i] would yield digital (84, 243, 273)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3243, 273, F3, 159) (dual of [273, 30, 160]-code), but
- residual code [i] would yield OA(384, 113, S3, 53), but
- the linear programming bound shows that M ≥ 220 344254 536683 114597 471552 228690 132437 887421 343564 925242 / 18336 783241 265797 > 384 [i]
- residual code [i] would yield OA(384, 113, S3, 53), but
- extracting embedded orthogonal array [i] would yield linear OA(3243, 273, F3, 159) (dual of [273, 30, 160]-code), but
(84, 84+160, 362)-Net in Base 3 — Upper bound on s
There is no (84, 244, 363)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 293 343017 805224 799774 545220 869033 771984 272568 698109 585917 348600 415419 480314 441257 353260 783168 244505 477989 404641 795937 > 3244 [i]