Best Known (57, 161, s)-Nets in Base 3
(57, 161, 48)-Net over F3 — Constructive and digital
Digital (57, 161, 48)-net over F3, using
- t-expansion [i] based on digital (45, 161, 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, 161, 64)-Net over F3 — Digital
Digital (57, 161, 64)-net over F3, using
- t-expansion [i] based on digital (49, 161, 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, 161, 231)-Net over F3 — Upper bound on s (digital)
There is no digital (57, 161, 232)-net over F3, because
- 2 times m-reduction [i] would yield digital (57, 159, 232)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3159, 232, F3, 102) (dual of [232, 73, 103]-code), but
- residual code [i] would yield OA(357, 129, S3, 34), but
- the linear programming bound shows that M ≥ 809354 275724 755763 388392 345759 739922 797610 964396 279375 665149 044386 576230 563429 788711 154133 451299 709022 738390 856898 439717 989850 127611 596351 869114 451031 438259 777221 098329 631772 962500 / 496 565899 544592 863342 975273 469710 153169 995856 683171 150954 405859 112820 872932 367752 443426 608748 461587 519331 493689 756263 492748 867872 843898 726516 425058 229629 > 357 [i]
- residual code [i] would yield OA(357, 129, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3159, 232, F3, 102) (dual of [232, 73, 103]-code), but
(57, 161, 255)-Net in Base 3 — Upper bound on s
There is no (57, 161, 256)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 73931 052519 525629 337282 462460 341491 517355 358722 917446 330443 050008 621543 290881 > 3161 [i]