Best Known (164−29, 164, s)-Nets in Base 4
(164−29, 164, 1268)-Net over F4 — Constructive and digital
Digital (135, 164, 1268)-net over F4, using
- 43 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
(164−29, 164, 15680)-Net over F4 — Digital
Digital (135, 164, 15680)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4164, 15680, F4, 29) (dual of [15680, 15516, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, 16442, F4, 29) (dual of [16442, 16278, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [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(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(416, 58, F4, 7) (dual of [58, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4164, 16442, F4, 29) (dual of [16442, 16278, 30]-code), using
(164−29, 164, large)-Net in Base 4 — Upper bound on s
There is no (135, 164, large)-net in base 4, because
- 27 times m-reduction [i] would yield (135, 137, large)-net in base 4, but