Best Known (171−109, 171, s)-Nets in Base 3
(171−109, 171, 48)-Net over F3 — Constructive and digital
Digital (62, 171, 48)-net over F3, using
- t-expansion [i] based on digital (45, 171, 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
(171−109, 171, 64)-Net over F3 — Digital
Digital (62, 171, 64)-net over F3, using
- t-expansion [i] based on digital (49, 171, 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
(171−109, 171, 259)-Net over F3 — Upper bound on s (digital)
There is no digital (62, 171, 260)-net over F3, because
- 1 times m-reduction [i] would yield digital (62, 170, 260)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3170, 260, F3, 108) (dual of [260, 90, 109]-code), but
- residual code [i] would yield OA(362, 151, S3, 36), but
- the linear programming bound shows that M ≥ 315 982382 936192 071390 140679 488499 072998 791757 288635 952439 440300 300796 107515 352026 429217 565172 043806 037893 027375 543805 938233 482903 869423 244973 363200 / 785 326280 415847 869963 489755 597656 207457 889325 536642 055726 952039 256677 666095 401973 721090 055553 083299 838666 281323 935199 > 362 [i]
- residual code [i] would yield OA(362, 151, S3, 36), but
- extracting embedded orthogonal array [i] would yield linear OA(3170, 260, F3, 108) (dual of [260, 90, 109]-code), but
(171−109, 171, 282)-Net in Base 3 — Upper bound on s
There is no (62, 171, 283)-net in base 3, because
- 1 times m-reduction [i] would yield (62, 170, 283)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1314 293527 045389 049709 359072 247478 528593 425160 290517 522710 557382 224726 517076 181549 > 3170 [i]