Best Known (57, 57+105, s)-Nets in Base 3
(57, 57+105, 48)-Net over F3 — Constructive and digital
Digital (57, 162, 48)-net over F3, using
- t-expansion [i] based on digital (45, 162, 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
(57, 57+105, 64)-Net over F3 — Digital
Digital (57, 162, 64)-net over F3, using
- t-expansion [i] based on digital (49, 162, 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
(57, 57+105, 216)-Net over F3 — Upper bound on s (digital)
There is no digital (57, 162, 217)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3162, 217, F3, 105) (dual of [217, 55, 106]-code), but
- residual code [i] would yield OA(357, 111, S3, 35), but
- the linear programming bound shows that M ≥ 37562 646628 132840 923555 510805 524990 698125 854470 702956 611458 206634 330592 972745 571811 693475 463280 794484 310358 798101 613598 520811 498715 852180 394100 485311 718464 678477 379766 250841 111527 263840 998708 126710 102901 010614 878078 058419 / 23 198198 819486 814111 531592 059961 917751 382231 381147 294006 713526 688083 077331 132750 604165 387990 072803 317950 890355 954749 807575 083764 923806 806712 425767 722092 751780 067760 482182 419810 536162 336560 924637 > 357 [i]
- residual code [i] would yield OA(357, 111, S3, 35), but
(57, 57+105, 255)-Net in Base 3 — Upper bound on s
There is no (57, 162, 256)-net in base 3, because
- 1 times m-reduction [i] would yield (57, 161, 256)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 73931 052519 525629 337282 462460 341491 517355 358722 917446 330443 050008 621543 290881 > 3161 [i]