Best Known (55, 55+95, s)-Nets in Base 3
(55, 55+95, 48)-Net over F3 — Constructive and digital
Digital (55, 150, 48)-net over F3, using
- t-expansion [i] based on digital (45, 150, 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+95, 64)-Net over F3 — Digital
Digital (55, 150, 64)-net over F3, using
- t-expansion [i] based on digital (49, 150, 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+95, 253)-Net over F3 — Upper bound on s (digital)
There is no digital (55, 150, 254)-net over F3, because
- 2 times m-reduction [i] would yield digital (55, 148, 254)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3148, 254, F3, 93) (dual of [254, 106, 94]-code), but
- residual code [i] would yield OA(355, 160, S3, 31), but
- 1 times truncation [i] would yield OA(354, 159, S3, 30), but
- the linear programming bound shows that M ≥ 57 512545 780273 008066 954941 653603 554048 685010 751297 100155 / 986015 082551 936588 824240 203061 > 354 [i]
- 1 times truncation [i] would yield OA(354, 159, S3, 30), but
- residual code [i] would yield OA(355, 160, S3, 31), but
- extracting embedded orthogonal array [i] would yield linear OA(3148, 254, F3, 93) (dual of [254, 106, 94]-code), but
(55, 55+95, 255)-Net in Base 3 — Upper bound on s
There is no (55, 150, 256)-net in base 3, because
- 1 times m-reduction [i] would yield (55, 149, 256)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 138556 416308 620569 874775 090118 014773 200031 213492 442904 433223 903473 814529 > 3149 [i]