Best Known (79, 79+143, s)-Nets in Base 3
(79, 79+143, 54)-Net over F3 — Constructive and digital
Digital (79, 222, 54)-net over F3, using
- net from sequence [i] based on digital (79, 53)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 53)-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, 53)-sequence over F9, using
(79, 79+143, 84)-Net over F3 — Digital
Digital (79, 222, 84)-net over F3, using
- t-expansion [i] based on digital (71, 222, 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
(79, 79+143, 337)-Net over F3 — Upper bound on s (digital)
There is no digital (79, 222, 338)-net over F3, because
- 2 times m-reduction [i] would yield digital (79, 220, 338)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3220, 338, F3, 141) (dual of [338, 118, 142]-code), but
- residual code [i] would yield OA(379, 196, S3, 47), but
- 5 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]
- 5 times truncation [i] would yield OA(374, 191, S3, 42), but
- residual code [i] would yield OA(379, 196, S3, 47), but
- extracting embedded orthogonal array [i] would yield linear OA(3220, 338, F3, 141) (dual of [338, 118, 142]-code), but
(79, 79+143, 350)-Net in Base 3 — Upper bound on s
There is no (79, 222, 351)-net in base 3, because
- 1 times m-reduction [i] would yield (79, 221, 351)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2947 258040 252535 364503 336651 645218 811129 484170 292484 390213 438691 159027 832624 733690 356105 182622 738145 103603 > 3221 [i]