Best Known (178−116, 178, s)-Nets in Base 3
(178−116, 178, 48)-Net over F3 — Constructive and digital
Digital (62, 178, 48)-net over F3, using
- t-expansion [i] based on digital (45, 178, 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
(178−116, 178, 64)-Net over F3 — Digital
Digital (62, 178, 64)-net over F3, using
- t-expansion [i] based on digital (49, 178, 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
(178−116, 178, 230)-Net over F3 — Upper bound on s (digital)
There is no digital (62, 178, 231)-net over F3, because
- 2 times m-reduction [i] would yield digital (62, 176, 231)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3176, 231, F3, 114) (dual of [231, 55, 115]-code), but
- residual code [i] would yield OA(362, 116, S3, 38), but
- the linear programming bound shows that M ≥ 317 092913 237312 931682 931230 145256 164570 899902 587673 477360 155450 374311 745951 419910 738271 470469 423748 300910 047030 562003 396664 131573 530146 865080 570074 280426 363280 161136 607519 358156 864069 602113 914689 631814 295536 292125 006549 / 788 684086 910466 147153 663101 284054 314401 233497 209874 715764 965309 878045 427152 638286 586748 473290 411470 012796 262989 930842 395578 057155 975611 958381 004404 206865 549402 120881 365315 438956 013078 947317 > 362 [i]
- residual code [i] would yield OA(362, 116, S3, 38), but
- extracting embedded orthogonal array [i] would yield linear OA(3176, 231, F3, 114) (dual of [231, 55, 115]-code), but
(178−116, 178, 273)-Net in Base 3 — Upper bound on s
There is no (62, 178, 274)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9 614246 464583 790311 922100 896400 186683 075990 220290 527799 694874 221774 947109 727506 857765 > 3178 [i]