Best Known (50, 145, s)-Nets in Base 3
(50, 145, 48)-Net over F3 — Constructive and digital
Digital (50, 145, 48)-net over F3, using
- t-expansion [i] based on digital (45, 145, 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
(50, 145, 64)-Net over F3 — Digital
Digital (50, 145, 64)-net over F3, using
- t-expansion [i] based on digital (49, 145, 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
(50, 145, 191)-Net over F3 — Upper bound on s (digital)
There is no digital (50, 145, 192)-net over F3, because
- 2 times m-reduction [i] would yield digital (50, 143, 192)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3143, 192, F3, 93) (dual of [192, 49, 94]-code), but
- residual code [i] would yield OA(350, 98, S3, 31), but
- the linear programming bound shows that M ≥ 10987 648641 013342 145743 173235 372682 266899 084370 827701 912518 996312 526775 125687 859494 241272 685867 404994 948584 725486 672070 814119 485618 889026 285759 139347 500994 272409 764144 107327 485143 / 14936 850860 728382 359380 496865 658716 917486 950067 604982 230925 851448 010473 789047 996877 004044 146341 975545 413487 942444 435229 603540 358524 720271 605160 064554 378035 > 350 [i]
- residual code [i] would yield OA(350, 98, S3, 31), but
- extracting embedded orthogonal array [i] would yield linear OA(3143, 192, F3, 93) (dual of [192, 49, 94]-code), but
(50, 145, 222)-Net in Base 3 — Upper bound on s
There is no (50, 145, 223)-net in base 3, because
- 1 times m-reduction [i] would yield (50, 144, 223)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 544 783955 701777 662867 701455 381768 399615 662646 489429 571766 073308 560163 > 3144 [i]