Best Known (234−152, 234, s)-Nets in Base 3
(234−152, 234, 57)-Net over F3 — Constructive and digital
Digital (82, 234, 57)-net over F3, using
- net from sequence [i] based on digital (82, 56)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 56)-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, 56)-sequence over F9, using
(234−152, 234, 84)-Net over F3 — Digital
Digital (82, 234, 84)-net over F3, using
- t-expansion [i] based on digital (71, 234, 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
(234−152, 234, 289)-Net over F3 — Upper bound on s (digital)
There is no digital (82, 234, 290)-net over F3, because
- 2 times m-reduction [i] would yield digital (82, 232, 290)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3232, 290, F3, 150) (dual of [290, 58, 151]-code), but
- residual code [i] would yield OA(382, 139, S3, 50), but
- the linear programming bound shows that M ≥ 1587 399042 866016 986303 890803 537138 245492 206166 258555 218457 073292 762725 146456 195326 962155 694534 950498 243860 513481 979544 181045 609868 510228 655317 898137 944968 273836 801278 213469 / 1 121662 775947 035202 501592 579753 339451 946215 941077 958998 362092 006927 786507 022495 318993 197038 312138 033878 088343 981406 194796 277022 130176 > 382 [i]
- residual code [i] would yield OA(382, 139, S3, 50), but
- extracting embedded orthogonal array [i] would yield linear OA(3232, 290, F3, 150) (dual of [290, 58, 151]-code), but
(234−152, 234, 358)-Net in Base 3 — Upper bound on s
There is no (82, 234, 359)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5061 277742 246203 857170 651858 951630 003330 696673 041767 424135 948406 844233 005632 768921 245603 232107 991543 731261 732969 > 3234 [i]