Best Known (53, 53+98, s)-Nets in Base 3
(53, 53+98, 48)-Net over F3 — Constructive and digital
Digital (53, 151, 48)-net over F3, using
- t-expansion [i] based on digital (45, 151, 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
(53, 53+98, 64)-Net over F3 — Digital
Digital (53, 151, 64)-net over F3, using
- t-expansion [i] based on digital (49, 151, 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
(53, 53+98, 212)-Net over F3 — Upper bound on s (digital)
There is no digital (53, 151, 213)-net over F3, because
- 2 times m-reduction [i] would yield digital (53, 149, 213)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3149, 213, F3, 96) (dual of [213, 64, 97]-code), but
- residual code [i] would yield OA(353, 116, S3, 32), but
- the linear programming bound shows that M ≥ 30 210723 713137 067177 793975 595437 389937 655484 598142 095463 860309 406988 954987 805424 205733 738518 195687 105322 605711 836142 015912 625489 745728 349344 380080 374822 240851 452346 970066 591441 / 1 515367 982730 495420 433932 584698 701039 350523 187407 082539 706877 207417 526512 266220 467507 257976 353777 430603 205351 330224 160963 457142 806942 300281 987731 316143 > 353 [i]
- residual code [i] would yield OA(353, 116, S3, 32), but
- extracting embedded orthogonal array [i] would yield linear OA(3149, 213, F3, 96) (dual of [213, 64, 97]-code), but
(53, 53+98, 236)-Net in Base 3 — Upper bound on s
There is no (53, 151, 237)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 128877 757841 838374 722046 157583 420444 776609 496505 180086 187719 095291 808283 > 3151 [i]