Best Known (171−111, 171, s)-Nets in Base 3
(171−111, 171, 48)-Net over F3 — Constructive and digital
Digital (60, 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
(171−111, 171, 64)-Net over F3 — Digital
Digital (60, 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
(171−111, 171, 222)-Net over F3 — Upper bound on s (digital)
There is no digital (60, 171, 223)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3171, 223, F3, 111) (dual of [223, 52, 112]-code), but
- residual code [i] would yield OA(360, 111, S3, 37), but
- the linear programming bound shows that M ≥ 123935 568792 005716 238073 093821 377566 373899 666515 891473 866546 732156 434237 662649 577761 952104 930600 687383 548328 236010 989613 216659 289681 775283 697533 382761 730971 562628 071493 118252 848986 661298 792237 / 2 651912 168878 349984 211244 004413 173913 645691 416562 954806 458129 553495 637719 806457 127468 273229 292776 586222 473938 179154 264603 376471 378754 376356 504885 590718 177008 802360 > 360 [i]
- residual code [i] would yield OA(360, 111, S3, 37), but
(171−111, 171, 267)-Net in Base 3 — Upper bound on s
There is no (60, 171, 268)-net in base 3, because
- 1 times m-reduction [i] would yield (60, 170, 268)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1436 609290 942301 059529 514232 716663 162930 620641 208256 949310 257619 401738 406167 939281 > 3170 [i]