Best Known (64, 64+116, s)-Nets in Base 3
(64, 64+116, 48)-Net over F3 — Constructive and digital
Digital (64, 180, 48)-net over F3, using
- t-expansion [i] based on digital (45, 180, 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
(64, 64+116, 64)-Net over F3 — Digital
Digital (64, 180, 64)-net over F3, using
- t-expansion [i] based on digital (49, 180, 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
(64, 64+116, 256)-Net over F3 — Upper bound on s (digital)
There is no digital (64, 180, 257)-net over F3, because
- 2 times m-reduction [i] would yield digital (64, 178, 257)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3178, 257, F3, 114) (dual of [257, 79, 115]-code), but
- residual code [i] would yield OA(364, 142, S3, 38), but
- the linear programming bound shows that M ≥ 2846 926043 960650 495051 068515 031396 348457 468672 309930 533525 992494 049197 887626 959665 996392 018123 051568 606408 359969 141109 818434 716934 446451 971089 981651 105587 855809 085244 860263 569249 657907 474743 124647 019708 720445 049205 956233 001230 173765 992166 211584 / 760 430386 438996 947207 913818 687184 087000 934370 497726 262841 427228 927142 510692 279719 900376 732452 532453 744502 189058 083069 971557 640736 908354 474342 397263 004008 984480 127369 741954 835724 634885 413693 063784 405933 734899 188533 > 364 [i]
- residual code [i] would yield OA(364, 142, S3, 38), but
- extracting embedded orthogonal array [i] would yield linear OA(3178, 257, F3, 114) (dual of [257, 79, 115]-code), but
(64, 64+116, 285)-Net in Base 3 — Upper bound on s
There is no (64, 180, 286)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 79 121731 157984 249625 908197 205173 430404 181235 138330 249238 683484 147088 584033 864202 216109 > 3180 [i]