Best Known (55, 55+103, s)-Nets in Base 3
(55, 55+103, 48)-Net over F3 — Constructive and digital
Digital (55, 158, 48)-net over F3, using
- t-expansion [i] based on digital (45, 158, 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
(55, 55+103, 64)-Net over F3 — Digital
Digital (55, 158, 64)-net over F3, using
- t-expansion [i] based on digital (49, 158, 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
(55, 55+103, 206)-Net over F3 — Upper bound on s (digital)
There is no digital (55, 158, 207)-net over F3, because
- 1 times m-reduction [i] would yield digital (55, 157, 207)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3157, 207, F3, 102) (dual of [207, 50, 103]-code), but
- residual code [i] would yield OA(355, 104, S3, 34), but
- the linear programming bound shows that M ≥ 156886 010858 484807 854054 378468 889712 644211 339363 001341 126789 276510 956457 437793 701769 100721 300683 454940 318668 109559 161920 994946 197728 957769 811558 637515 615510 901840 516204 119571 / 885 396566 461856 638142 869755 626651 841542 071726 372464 475427 680706 478414 432865 627906 503409 309515 904907 562198 156244 985091 365409 869297 751679 479371 690668 > 355 [i]
- residual code [i] would yield OA(355, 104, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3157, 207, F3, 102) (dual of [207, 50, 103]-code), but
(55, 55+103, 244)-Net in Base 3 — Upper bound on s
There is no (55, 158, 245)-net in base 3, because
- 1 times m-reduction [i] would yield (55, 157, 245)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 811 523338 735210 755107 043197 901031 573322 558327 682283 271847 804316 372814 237931 > 3157 [i]