Best Known (163−29, 163, s)-Nets in Base 4
(163−29, 163, 1268)-Net over F4 — Constructive and digital
Digital (134, 163, 1268)-net over F4, using
- 42 times duplication [i] based on digital (132, 161, 1268)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (31, 45, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- digital (87, 116, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- digital (31, 45, 240)-net over F4, using
- (u, u+v)-construction [i] based on
(163−29, 163, 14894)-Net over F4 — Digital
Digital (134, 163, 14894)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4163, 14894, F4, 29) (dual of [14894, 14731, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 16434, F4, 29) (dual of [16434, 16271, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4161, 16432, F4, 29) (dual of [16432, 16271, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(413, 48, F4, 6) (dual of [48, 35, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4161, 16432, F4, 29) (dual of [16432, 16271, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 16434, F4, 29) (dual of [16434, 16271, 30]-code), using
(163−29, 163, large)-Net in Base 4 — Upper bound on s
There is no (134, 163, large)-net in base 4, because
- 27 times m-reduction [i] would yield (134, 136, large)-net in base 4, but