Best Known (49, 207, s)-Nets in Base 4
(49, 207, 66)-Net over F4 — Constructive and digital
Digital (49, 207, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second 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) = 49 and N(F) ≥ 66, using
(49, 207, 81)-Net over F4 — Digital
Digital (49, 207, 81)-net over F4, using
- t-expansion [i] based on digital (46, 207, 81)-net over F4, using
- net from sequence [i] based on digital (46, 80)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 46 and N(F) ≥ 81, using
- net from sequence [i] based on digital (46, 80)-sequence over F4, using
(49, 207, 204)-Net over F4 — Upper bound on s (digital)
There is no digital (49, 207, 205)-net over F4, because
- 10 times m-reduction [i] would yield digital (49, 197, 205)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4197, 205, F4, 148) (dual of [205, 8, 149]-code), but
- construction Y1 [i] would yield
- linear OA(4196, 201, F4, 148) (dual of [201, 5, 149]-code), but
- residual code [i] would yield linear OA(448, 52, F4, 37) (dual of [52, 4, 38]-code), but
- 1 times truncation [i] would yield linear OA(447, 51, F4, 36) (dual of [51, 4, 37]-code), but
- residual code [i] would yield linear OA(448, 52, F4, 37) (dual of [52, 4, 38]-code), but
- OA(48, 205, S4, 4), but
- discarding factors would yield OA(48, 121, S4, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 65704 > 48 [i]
- discarding factors would yield OA(48, 121, S4, 4), but
- linear OA(4196, 201, F4, 148) (dual of [201, 5, 149]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4197, 205, F4, 148) (dual of [205, 8, 149]-code), but
(49, 207, 210)-Net in Base 4 — Upper bound on s
There is no (49, 207, 211)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(4207, 211, S4, 158), but
- the (dual) Plotkin bound shows that M ≥ 2 707685 248164 858261 307045 101702 230179 137145 581421 695874 189921 465443 966120 903931 272499 975005 961073 806735 733604 454495 675614 232576 / 53 > 4207 [i]