Best Known (157−117, 157, s)-Nets in Base 4
(157−117, 157, 56)-Net over F4 — Constructive and digital
Digital (40, 157, 56)-net over F4, using
- t-expansion [i] based on digital (33, 157, 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
(157−117, 157, 75)-Net over F4 — Digital
Digital (40, 157, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
(157−117, 157, 189)-Net over F4 — Upper bound on s (digital)
There is no digital (40, 157, 190)-net over F4, because
- 1 times m-reduction [i] would yield digital (40, 156, 190)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4156, 190, F4, 116) (dual of [190, 34, 117]-code), but
- construction Y1 [i] would yield
- linear OA(4155, 170, F4, 116) (dual of [170, 15, 117]-code), but
- construction Y1 [i] would yield
- linear OA(4154, 162, F4, 116) (dual of [162, 8, 117]-code), but
- construction Y1 [i] would yield
- linear OA(4153, 158, F4, 116) (dual of [158, 5, 117]-code), but
- residual code [i] would yield linear OA(437, 41, F4, 29) (dual of [41, 4, 30]-code), but
- 1 times truncation [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
- 1 times truncation [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- residual code [i] would yield linear OA(437, 41, F4, 29) (dual of [41, 4, 30]-code), but
- OA(48, 162, 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(4153, 158, F4, 116) (dual of [158, 5, 117]-code), but
- construction Y1 [i] would yield
- OA(415, 170, 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(4154, 162, F4, 116) (dual of [162, 8, 117]-code), but
- construction Y1 [i] would yield
- OA(434, 190, S4, 20), but
- discarding factors would yield OA(434, 173, S4, 20), but
- the Rao or (dual) Hamming bound shows that M ≥ 305 996090 287239 486286 > 434 [i]
- discarding factors would yield OA(434, 173, S4, 20), but
- linear OA(4155, 170, F4, 116) (dual of [170, 15, 117]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4156, 190, F4, 116) (dual of [190, 34, 117]-code), but
(157−117, 157, 266)-Net in Base 4 — Upper bound on s
There is no (40, 157, 267)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 156, 267)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 9592 192330 021642 682090 208449 965863 531809 813064 796898 198517 222430 004502 281809 474998 623664 801904 > 4156 [i]