Best Known (39, 220, s)-Nets in Base 4
(39, 220, 56)-Net over F4 — Constructive and digital
Digital (39, 220, 56)-net over F4, using
- t-expansion [i] based on digital (33, 220, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(39, 220, 66)-Net over F4 — Digital
Digital (39, 220, 66)-net over F4, using
- t-expansion [i] based on digital (37, 220, 66)-net over F4, using
- net from sequence [i] based on digital (37, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 37 and N(F) ≥ 66, using
- net from sequence [i] based on digital (37, 65)-sequence over F4, using
(39, 220, 165)-Net over F4 — Upper bound on s (digital)
There is no digital (39, 220, 166)-net over F4, because
- 61 times m-reduction [i] would yield digital (39, 159, 166)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4159, 166, F4, 120) (dual of [166, 7, 121]-code), but
- residual code [i] would yield linear OA(439, 45, F4, 30) (dual of [45, 6, 31]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(4159, 166, F4, 120) (dual of [166, 7, 121]-code), but
(39, 220, 170)-Net in Base 4 — Upper bound on s
There is no (39, 220, 171)-net in base 4, because
- 53 times m-reduction [i] would yield (39, 167, 171)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4167, 171, S4, 128), but
- the (dual) Plotkin bound shows that M ≥ 2 239744 742177 804210 557442 280568 444278 121645 497234 649534 899989 100963 791871 180160 945380 877493 271607 115776 / 43 > 4167 [i]
- extracting embedded orthogonal array [i] would yield OA(4167, 171, S4, 128), but