Best Known (38, 38+114, s)-Nets in Base 4
(38, 38+114, 56)-Net over F4 — Constructive and digital
Digital (38, 152, 56)-net over F4, using
- t-expansion [i] based on digital (33, 152, 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
(38, 38+114, 66)-Net over F4 — Digital
Digital (38, 152, 66)-net over F4, using
- t-expansion [i] based on digital (37, 152, 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
(38, 38+114, 164)-Net over F4 — Upper bound on s (digital)
There is no digital (38, 152, 165)-net over F4, because
- 2 times m-reduction [i] would yield digital (38, 150, 165)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4150, 165, F4, 112) (dual of [165, 15, 113]-code), but
- construction Y1 [i] would yield
- linear OA(4149, 157, F4, 112) (dual of [157, 8, 113]-code), but
- construction Y1 [i] would yield
- linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- residual code [i] would yield linear OA(48, 11, F4, 7) (dual of [11, 3, 8]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- OA(48, 157, 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(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- construction Y1 [i] would yield
- OA(415, 165, S4, 8), but
- discarding factors would yield OA(415, 135, S4, 8), but
- the Rao or (dual) Hamming bound shows that M ≥ 1082 768311 > 415 [i]
- discarding factors would yield OA(415, 135, S4, 8), but
- linear OA(4149, 157, F4, 112) (dual of [157, 8, 113]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4150, 165, F4, 112) (dual of [165, 15, 113]-code), but
(38, 38+114, 252)-Net in Base 4 — Upper bound on s
There is no (38, 152, 253)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 36 842846 982321 153589 998720 629171 418196 210949 866828 621862 388718 324867 392658 083924 720891 792592 > 4152 [i]