Best Known (53, 53+101, s)-Nets in Base 3
(53, 53+101, 48)-Net over F3 — Constructive and digital
Digital (53, 154, 48)-net over F3, using
- t-expansion [i] based on digital (45, 154, 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
(53, 53+101, 64)-Net over F3 — Digital
Digital (53, 154, 64)-net over F3, using
- t-expansion [i] based on digital (49, 154, 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
(53, 53+101, 197)-Net over F3 — Upper bound on s (digital)
There is no digital (53, 154, 198)-net over F3, because
- 2 times m-reduction [i] would yield digital (53, 152, 198)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3152, 198, F3, 99) (dual of [198, 46, 100]-code), but
- residual code [i] would yield OA(353, 98, S3, 33), but
- the linear programming bound shows that M ≥ 70405 986787 631825 456516 749181 938054 391496 023445 621407 247393 974860 100167 193867 925850 156399 787754 027088 012405 632448 512652 067196 819333 / 3435 057828 133240 896934 525314 441640 529877 282946 429987 863449 445479 386348 848486 854894 335014 985183 953870 063104 > 353 [i]
- residual code [i] would yield OA(353, 98, S3, 33), but
- extracting embedded orthogonal array [i] would yield linear OA(3152, 198, F3, 99) (dual of [198, 46, 100]-code), but
(53, 53+101, 234)-Net in Base 3 — Upper bound on s
There is no (53, 154, 235)-net in base 3, because
- 1 times m-reduction [i] would yield (53, 153, 235)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10 415075 276235 816390 215762 985803 340606 958004 818095 119051 064683 766379 311381 > 3153 [i]