Best Known (174−117, 174, s)-Nets in Base 3
(174−117, 174, 48)-Net over F3 — Constructive and digital
Digital (57, 174, 48)-net over F3, using
- t-expansion [i] based on digital (45, 174, 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
(174−117, 174, 64)-Net over F3 — Digital
Digital (57, 174, 64)-net over F3, using
- t-expansion [i] based on digital (49, 174, 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
(174−117, 174, 179)-Net over F3 — Upper bound on s (digital)
There is no digital (57, 174, 180)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3174, 180, F3, 117) (dual of [180, 6, 118]-code), but
- residual code [i] would yield linear OA(357, 62, F3, 39) (dual of [62, 5, 40]-code), but
- residual code [i] would yield linear OA(318, 22, F3, 13) (dual of [22, 4, 14]-code), but
- 1 times truncation [i] would yield linear OA(317, 21, F3, 12) (dual of [21, 4, 13]-code), but
- residual code [i] would yield linear OA(318, 22, F3, 13) (dual of [22, 4, 14]-code), but
- residual code [i] would yield linear OA(357, 62, F3, 39) (dual of [62, 5, 40]-code), but
(174−117, 174, 244)-Net in Base 3 — Upper bound on s
There is no (57, 174, 245)-net in base 3, because
- 1 times m-reduction [i] would yield (57, 173, 245)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 41644 317570 530993 909684 840828 795442 749086 457814 623517 823213 138296 978171 569280 297489 > 3173 [i]