Best Known (173−112, 173, s)-Nets in Base 3
(173−112, 173, 48)-Net over F3 — Constructive and digital
Digital (61, 173, 48)-net over F3, using
- t-expansion [i] based on digital (45, 173, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(173−112, 173, 64)-Net over F3 — Digital
Digital (61, 173, 64)-net over F3, using
- t-expansion [i] based on digital (49, 173, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(173−112, 173, 234)-Net over F3 — Upper bound on s (digital)
There is no digital (61, 173, 235)-net over F3, because
- 1 times m-reduction [i] would yield digital (61, 172, 235)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3172, 235, F3, 111) (dual of [235, 63, 112]-code), but
- residual code [i] would yield OA(361, 123, S3, 37), but
- the linear programming bound shows that M ≥ 53802 787032 693826 686905 780400 028347 371541 329262 674589 454541 142212 262937 890911 311162 908807 748436 909454 573027 275662 890346 508432 847103 298034 682895 774016 561066 435642 461929 340637 119224 665789 434953 009143 441226 431234 826813 100819 098036 674839 090749 169691 593728 264212 683283 / 418605 786084 212811 158552 747152 843863 000680 691386 758976 397047 999955 369991 488044 718554 364682 919940 478858 548806 967884 215208 712362 889491 710739 004666 558992 783423 040936 064833 282715 648854 777487 983164 505778 609794 477464 440492 904435 881907 396168 > 361 [i]
- residual code [i] would yield OA(361, 123, S3, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(3172, 235, F3, 111) (dual of [235, 63, 112]-code), but
(173−112, 173, 271)-Net in Base 3 — Upper bound on s
There is no (61, 173, 272)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 38602 074576 453491 868187 723627 677269 245968 419406 267818 685512 615392 629419 867589 899265 > 3173 [i]