Best Known (58, 159, s)-Nets in Base 3
(58, 159, 48)-Net over F3 — Constructive and digital
Digital (58, 159, 48)-net over F3, using
- t-expansion [i] based on digital (45, 159, 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
(58, 159, 64)-Net over F3 — Digital
Digital (58, 159, 64)-net over F3, using
- t-expansion [i] based on digital (49, 159, 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
(58, 159, 261)-Net over F3 — Upper bound on s (digital)
There is no digital (58, 159, 262)-net over F3, because
- 2 times m-reduction [i] would yield digital (58, 157, 262)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3157, 262, F3, 99) (dual of [262, 105, 100]-code), but
- residual code [i] would yield OA(358, 162, S3, 33), but
- 1 times truncation [i] would yield OA(357, 161, S3, 32), but
- the linear programming bound shows that M ≥ 721213 722132 726616 769951 419548 792269 427090 005182 815014 600979 / 455 415925 085809 684165 452127 356779 > 357 [i]
- 1 times truncation [i] would yield OA(357, 161, S3, 32), but
- residual code [i] would yield OA(358, 162, S3, 33), but
- extracting embedded orthogonal array [i] would yield linear OA(3157, 262, F3, 99) (dual of [262, 105, 100]-code), but
(58, 159, 267)-Net in Base 3 — Upper bound on s
There is no (58, 159, 268)-net in base 3, because
- 1 times m-reduction [i] would yield (58, 158, 268)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2817 418915 487455 084956 314112 165720 309024 018223 493996 563857 808248 412186 166889 > 3158 [i]