Notation

Symbol	Meaning
\(\mathcal{K}_{i_b}^{1}\)	the set of indices whose bit \(i_b\) equals one; \(\mathcal{K}_{i_b}^{0}\) is its complement

1. Gray code

1.1 In-phase Plane

The Gray code \(g\) of the binary index \(b\) is,

\[g = b \oplus \left\lfloor \frac{b}{2} \right\rfloor\]

\(\bigoplus\) is the bitwise exclusive or and \(\lfloor \cdot \rfloor\) is the floor. The in-phase modulated signal is

\[x_{\mathrm{I}}(b) = 2b + 1 - 2^{N_{\mathrm{bits}}}\]

The Gray code examples are listed below.

\(N_{\mathrm{bits}}\)	bits \(g\)	\(x_{\mathrm{I}}(b)\)
\(1\)	\(0, 1\)	\(-1, +1\)
\(2\)	\(0, 1, 3, 2\)	\(-3, -1, +1, +3\)
\(3\)	\(0, 1, 3, 2, 6, 7, 5, 4\)	\(-7, -5, -3, -1, +1, +3, +5, +7\)
\(4\)	\(0, 1, 3, 2, 6, 7, 5, 4\) \(12, 13, 15, 14, 10, 11, 9, 8\)	\(-15, -13, -11, -9, -7, -5, -3, -1\) \(+1, +3, +5, +7, +9, +11, +13, +15\)
\(5\)	\(0, 1, 3, 2, 6, 7, 5, 4\) \(12, 13, 15, 14, 10, 11, 9, 8\) \(24, 25, 27, 26, 30, 31, 29, 28\) \(20, 21, 23, 22, 18, 19, 17, 16\)	\(-31, -29, -27, -25, -23, -21, -19, -17\) \(-15, -13, -11, -9, -7, -5, -3, -1\) \(+1, +3, +5, +7, +9, +11, +13, +15\) \(+17, +19, +21, +23, +25, +27, +29, +31\)
\(6\)	\(0, 1, 3, 2, 6, 7, 5, 4\) \(12, 13, 15, 14, 10, 11, 9, 8\) \(24, 25, 27, 26, 30, 31, 29, 28\) \(20, 21, 23, 22, 18, 19, 17, 16\) \(48, 49, 51, 50, 54, 55, 53, 52\) \(60, 61, 63, 62, 58, 59, 57, 56\) \(40, 41, 43, 42, 46, 47, 45, 44\) \(36, 37, 39, 38, 34, 35, 33, 32\)	\(-63, -61, -59, -57, -55, -53, -51, -49\) \(-47, -45, -43, -41, -39, -37, -35, -33\) \(-31, -29, -27, -25, -23, -21, -19, -17\) \(-15, -13, -11, -9, -7, -5, -3, -1\) \(+1, +3, +5, +7, +9, +11, +13, +15\) \(+17, +19, +21, +23, +25, +27, +29, +31\) \(+33, +35, +37, +39, +41, +43, +45, +47\) \(+49, +51, +53, +55, +57, +59, +61, +63\)

1.2 Complex Plane

The normalised modulated signal in the complex plane is

\[x = A \left( x_{\mathrm{I}}(b_{\mathrm{I}}) + j\, x_{\mathrm{Q}}(b_{\mathrm{Q}}) \right)\]

where \(b_{\mathrm{I}}\) and \(b_{\mathrm{Q}}\) are the indices of the two axes. The bits are split by axis: the in-phase axis carries the first half, \(g_0, \dots, g_{N_{\mathrm{bits}}-1}\), and the quadrature axis the second half, \(g_{N_{\mathrm{bits}}}, \dots, g_{2N_{\mathrm{bits}}-1}\).

The scale factor \(A\) is fixed to give unit average power.

\[\mathbb{E}\left[ \lvert x \rvert^{2} \right] = 2\, A^{2}\, \mathbb{E}\left[ g^{2} \right]\] \[\mathbb{E}\left[ g^{2} \right] = \frac{1}{2^{N_{\mathrm{bits}}}} \sum_{k=0}^{2^{N_{\mathrm{bits}}}-1} \left( 2k + 1 - 2^{N_{\mathrm{bits}}} \right)^{2} = \frac{1^{2} + 3^{2} + \dots + \left( 2^{N_{\mathrm{bits}}}-1 \right)^{2}}{2^{N_{\mathrm{bits}}-1}} = \frac{2^{2N_{\mathrm{bits}}} - 1}{3}\]

Consider both the I/Q plane, unit energy requires

\[A^{2} = \frac{3}{2\left( 2^{2N_{\mathrm{bits}}} - 1 \right)}\]

We define

\[x_{\mathrm{norm,I}} = A\, x_{\mathrm{I}}(b_{\mathrm{I}}), \qquad x_{\mathrm{norm,Q}} = A\, x_{\mathrm{Q}}(b_{\mathrm{Q}})\]

Without loss of generality, every expression below is written for the in-phase signal.

2. LLR Definition

The demapper sees one real branch observation with an additive white Gaussian impairment,

\[\hat{x}_{\mathrm{norm,I}} = x_{\mathrm{norm,I}} + n, \qquad n \sim \mathcal{N}\left( 0, \frac{\sigma^2}{2} \right)\]

The symbol is normalised to unit average power (i.e., \(\mathbb{E}\left[ \lvert x \rvert^2 \right] = 1\)), so each branch carries half of it, \(\mathbb{E}\left[ x_{\mathrm{norm,I}}^2 \right] = 1/2\). The noise power \(\sigma^2\) is split in the same way. The signal-to-noise ratio of the branch is therefore

\[\frac{\mathbb{E}\left[ x_{\mathrm{norm,I}}^2 \right]}{\mathbb{E}\left[ n^2 \right]} = \frac{1/2}{\sigma^2/2} = \frac{1}{\sigma^2}\]

The density for \(x_{\mathrm{I}}\) is

\[p(\hat{x}_{\mathrm{I}} \mid x_{\mathrm{I}}) = \frac{1}{\sqrt{\pi\, \sigma_A^{2}}} \exp\left( -\frac{(\hat{x}_{\mathrm{I}} - x_{\mathrm{I}})^2}{\sigma_A^{2}} \right)\]

where the noise power is \(\sigma_A^{2} = \sigma^{2}/A^{2}\).

The LLR is

\[\begin{gathered} \mathrm{LLR}_{i_b} = \ln \frac{P\left( g_{i_b} = 1 \mid \hat{x}_{\mathrm{I}} \right)}{P\left( g_{i_b} = 0 \mid \hat{x}_{\mathrm{I}} \right)} \\ \mathrm{LLR}_{i_b} = \ln \sum_{b \in \mathcal{K}_{i_b}^1} e^{-(\hat{x}_{\mathrm{I}} - x_{\mathrm{I}}(b))^2/\sigma_A^{2}} - \ln \sum_{b \in \mathcal{K}_{i_b}^0} e^{-(\hat{x}_{\mathrm{I}} - x_{\mathrm{I}}(b))^2/\sigma_A^{2}} \end{gathered}\]

where \(x_{\mathrm{I}}(b) = 2b + 1 - 2^{N_{\mathrm{bits}}}\) is the value of the point with index \(b\) in units of \(A\), \(\mathcal{K}_{i_b}^{1} = \left\{ b : g_{i_b} = 1 \right\}\) collects the indices whose bit \(i_b\) equals one, \(\mathcal{K}_{i_b}^{0}\) is its complement.

2.1 LLR of BPSK and QPSK

\[\mathrm{LLR}_0 = \ln e^{-(\hat{x}_{\mathrm{I}} - 1)^2/\sigma_A^{2}} - \ln e^{-(\hat{x}_{\mathrm{I}} + 1)^2/\sigma_A^{2}}\] \[\mathrm{LLR}_0 = \frac{4 \hat{x}_{\mathrm{I}}}{\sigma_A^{2}}\]

The quadrature bit of QPSK is the same expression with \(\hat{x}_{\mathrm{Q}}\) in place of \(\hat{x}_{\mathrm{I}}\).

2.2 LLR of 16QAM

Gray code \(g_{\mathrm{I}}\)	\(x_{\mathrm{I}}\)
\(00\)	\(-3\)
\(01\)	\(-1\)
\(11\)	\(+1\)
\(10\)	\(+3\)

\[\begin{cases} \mathrm{LLR}_0 = \ln \left( e^{-(\hat{x}_{\mathrm{I}} - 1)^2/\sigma_A^{2}} + e^{-(\hat{x}_{\mathrm{I}} - 3)^2/\sigma_A^{2}} \right) - \ln \left( e^{-(\hat{x}_{\mathrm{I}} + 1)^2/\sigma_A^{2}} + e^{-(\hat{x}_{\mathrm{I}} + 3)^2/\sigma_A^{2}} \right) \\ \mathrm{LLR}_1 = \ln \left( e^{-(\hat{x}_{\mathrm{I}} + 1)^2/\sigma_A^{2}} + e^{-(\hat{x}_{\mathrm{I}} - 1)^2/\sigma_A^{2}} \right) - \ln \left( e^{-(\hat{x}_{\mathrm{I}} + 3)^2/\sigma_A^{2}} + e^{-(\hat{x}_{\mathrm{I}} - 3)^2/\sigma_A^{2}} \right) \end{cases}\]

3. Max-Log LLR

Replacing the log-sum-exp by its largest term,

\[\ln \sum_{k} e^{a_k} \approx \max_{k} a_k\]

The max-log LLR is

\[\mathrm{LLR}_{i_b} = \frac{1}{\sigma_A^{2}} \left[ \min_{b \in \mathcal{K}_{i_b}^0} \left( \hat{x}_{\mathrm{I}} - x_{\mathrm{I}}(b) \right)^2 - \min_{b \in \mathcal{K}_{i_b}^1} \left( \hat{x}_{\mathrm{I}} - x_{\mathrm{I}}(b) \right)^2 \right]\]

Denoting by \(x_{\mathrm{I},\min}^{(1)}\) and \(x_{\mathrm{I},\min}^{(0)}\) the two nearest values with bit 1 and bit 0,

\[\begin{aligned} \mathrm{LLR}_{i_b} &= \frac{1}{\sigma_A^{2}} \left[ \left( \hat{x}_{\mathrm{I}} - x_{\mathrm{I},\min}^{(0)} \right)^2 - \left( \hat{x}_{\mathrm{I}} - x_{\mathrm{I},\min}^{(1)} \right)^2 \right] \\ &= \frac{2 \left( x_{\mathrm{I},\min}^{(1)} - x_{\mathrm{I},\min}^{(0)} \right)}{\sigma_A^{2}} \left( \hat{x}_{\mathrm{I}} - \frac{x_{\mathrm{I},\min}^{(1)} + x_{\mathrm{I},\min}^{(0)}}{2} \right) \end{aligned}\]

3.1 BPSK/QPSK

For \(i_b = 0\), with \(\hat{x}_{\mathrm{I}} > 0\),

\[\left( x_{\mathrm{I},\min}^{(1)},\ x_{\mathrm{I},\min}^{(0)} \right) = (+1,\ -1)\] \[\mathrm{LLR}_{i_b} = \frac{4\, \hat{x}_{\mathrm{I}}}{\sigma_A^{2}}\]

3.2 16QAM

For \(i_b = 0\), with \(\hat{x}_{\mathrm{I}} > 0\),

\[\left( x_{\mathrm{I},\min}^{(1)},\ x_{\mathrm{I},\min}^{(0)} \right) = \begin{cases} \left( +1,\ -1 \right), & \hat{x}_{\mathrm{I}} \le 2 \\ \left( +3,\ -1 \right), & \hat{x}_{\mathrm{I}} > 2 \end{cases}\] \[\mathrm{LLR}_{i_b} = \begin{cases} \frac{4\, \hat{x}_{\mathrm{I}}}{\sigma_A^{2}}, & \hat{x}_{\mathrm{I}} \le 2 \\ \frac{8 \left( \hat{x}_{\mathrm{I}} - 1 \right)}{\sigma_A^{2}}, & \hat{x}_{\mathrm{I}} > 2 \end{cases}\]

For \(i_b = 1\),

\[\left( x_{\mathrm{I},\min}^{(1)},\ x_{\mathrm{I},\min}^{(0)} \right) = (+1,\ +3)\] \[\mathrm{LLR}_{i_b} = \frac{4 \left( 2 - \lvert \hat{x}_{\mathrm{I}} \rvert \right)}{\sigma_A^{2}}\]

3.3 64QAM

With the sample on the positive side, each in-phase bit takes the nearest value of each class,

\(i_b\)	\(x_{\mathrm{I},\min}\)	\(\hat{x}_{\mathrm{I}}\)
\(0\)	\(\left\{ +1,\ -1 \right\}\)	\((0,\ 2]\)
\(0\)	\(\left\{ +3,\ -1 \right\}\)	\((2,\ 4]\)
\(0\)	\(\left\{ +5,\ -1 \right\}\)	\((4,\ 6]\)
\(0\)	\(\left\{ +7,\ -1 \right\}\)	\((6,\ \infty)\)
\(1\)	\(\left\{ +1,\ +5 \right\}\)	\((0,\ 2]\)
\(1\)	\(\left\{ +3,\ +5 \right\}\)	\((2,\ 6]\)
\(1\)	\(\left\{ +3,\ +7 \right\}\)	\((6,\ \infty)\)
\(2\)	\(\left\{ +3,\ +1 \right\}\)	\((0,\ 4]\)
\(2\)	\(\left\{ +5,\ +7 \right\}\)	\((4,\ \infty)\)

3.4 Recursion

The recursion approach is introduced in Ref. [1].

\[g_0 = \hat{x}_{\mathrm{I}}\] \[g_{i_b+1} = -\left| g_{i_b} \right| + 2^{\,N_{\mathrm{bits}}-1-i_b}\] \[\mathrm{LLR}_{i_b} = \frac{4}{\sigma_A^{2}}\, g_{i_b}\]

4. The LLR of Multiple Streams

In maximum-likelihood detection, the observation is the vector \(\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}\) of \(N_r\) complex samples, where \(\mathbf{n} \sim \mathcal{CN}\left( \mathbf{0}, \sigma^{2}\mathbf{I} \right)\). Here \(\mathbf{H}\) has dimensions \(N_r \times N_t\). The density is the joint likelihood of the whole received vector,

\[p(\mathbf{y} \mid \mathbf{x}) = \frac{1}{\left( \pi \sigma^{2} \right)^{N_r}} \exp\left( -\frac{D(\mathbf{x})}{\sigma^{2}} \right)\]

where

\[D(\mathbf{x}) = \left\lVert \mathbf{y} - \mathbf{H} \mathbf{x} \right\rVert^{2}\]

The LLR is

\[\mathrm{LLR}_{i_b} = \ln \sum_{\mathbf{x} \in \mathcal{X}_{i_b}^{1}} e^{-D(\mathbf{x})/\sigma^{2}} - \ln \sum_{\mathbf{x} \in \mathcal{X}_{i_b}^{0}} e^{-D(\mathbf{x})/\sigma^{2}}\]

The max-log LLR is

\[\mathrm{LLR}_{i_b} = \frac{1}{\sigma^{2}} \left[ \min_{\mathbf{x} \in \mathcal{X}_{i_b}^{0}} D(\mathbf{x}) - \min_{\mathbf{x} \in \mathcal{X}_{i_b}^{1}} D(\mathbf{x}) \right]\]

References

[1] K. S. Kim, K. M. Hyun, D. Yoon, C. W. Yu, Y. O. Park, S. K. Park, “General Log-Likelihood Ratio Expression and Its Implementation Algorithm for Gray-Coded QAM Signals,” ETRI Journal, vol. 28, no. 3, pp. 291-300, 2006.

Appendix A.

A.1 256QAM

\(i_b\)	\(x_{\mathrm{I},\min}\)	\(\hat{x}_{\mathrm{I}}\)
\(0\)	\(\left\{ +1,\ -1 \right\}\)	\((0,\ 2]\)
\(0\)	\(\left\{ +3,\ -1 \right\}\)	\((2,\ 4]\)
\(0\)	\(\left\{ +5,\ -1 \right\}\)	\((4,\ 6]\)
\(0\)	\(\left\{ +7,\ -1 \right\}\)	\((6,\ 8]\)
\(0\)	\(\left\{ +9,\ -1 \right\}\)	\((8,\ 10]\)
\(0\)	\(\left\{ +11,\ -1 \right\}\)	\((10,\ 12]\)
\(0\)	\(\left\{ +13,\ -1 \right\}\)	\((12,\ 14]\)
\(0\)	\(\left\{ +15,\ -1 \right\}\)	\((14,\ \infty)\)
\(1\)	\(\left\{ +1,\ +9 \right\}\)	\((0,\ 2]\)
\(1\)	\(\left\{ +3,\ +9 \right\}\)	\((2,\ 4]\)
\(1\)	\(\left\{ +5,\ +9 \right\}\)	\((4,\ 6]\)
\(1\)	\(\left\{ +7,\ +9 \right\}\)	\((6,\ 10]\)
\(1\)	\(\left\{ +7,\ +11 \right\}\)	\((10,\ 12]\)
\(1\)	\(\left\{ +7,\ +13 \right\}\)	\((12,\ 14]\)
\(1\)	\(\left\{ +7,\ +15 \right\}\)	\((14,\ \infty)\)
\(2\)	\(\left\{ +5,\ +1 \right\}\)	\((0,\ 2]\)
\(2\)	\(\left\{ +5,\ +3 \right\}\)	\((2,\ 6]\)
\(2\)	\(\left\{ +7,\ +3 \right\}\)	\((6,\ 8]\)
\(2\)	\(\left\{ +9,\ +13 \right\}\)	\((8,\ 10]\)
\(2\)	\(\left\{ +11,\ +13 \right\}\)	\((10,\ 14]\)
\(2\)	\(\left\{ +11,\ +15 \right\}\)	\((14,\ \infty)\)
\(3\)	\(\left\{ +3,\ +1 \right\}\)	\((0,\ 4]\)
\(3\)	\(\left\{ +5,\ +7 \right\}\)	\((4,\ 8]\)
\(3\)	\(\left\{ +11,\ +9 \right\}\)	\((8,\ 12]\)
\(3\)	\(\left\{ +13,\ +15 \right\}\)	\((12,\ \infty)\)

A.2 1024QAM

\(i_b\)	\(x_{\mathrm{I},\min}\)	\(\hat{x}_{\mathrm{I}}\)
\(0\)	\(\left\{ +1,\ -1 \right\}\)	\((0,\ 2]\)
\(0\)	\(\left\{ +3,\ -1 \right\}\)	\((2,\ 4]\)
\(0\)	\(\left\{ +5,\ -1 \right\}\)	\((4,\ 6]\)
\(0\)	\(\left\{ +7,\ -1 \right\}\)	\((6,\ 8]\)
\(0\)	\(\left\{ +9,\ -1 \right\}\)	\((8,\ 10]\)
\(0\)	\(\left\{ +11,\ -1 \right\}\)	\((10,\ 12]\)
\(0\)	\(\left\{ +13,\ -1 \right\}\)	\((12,\ 14]\)
\(0\)	\(\left\{ +15,\ -1 \right\}\)	\((14,\ 16]\)
\(0\)	\(\left\{ +17,\ -1 \right\}\)	\((16,\ 18]\)
\(0\)	\(\left\{ +19,\ -1 \right\}\)	\((18,\ 20]\)
\(0\)	\(\left\{ +21,\ -1 \right\}\)	\((20,\ 22]\)
\(0\)	\(\left\{ +23,\ -1 \right\}\)	\((22,\ 24]\)
\(0\)	\(\left\{ +25,\ -1 \right\}\)	\((24,\ 26]\)
\(0\)	\(\left\{ +27,\ -1 \right\}\)	\((26,\ 28]\)
\(0\)	\(\left\{ +29,\ -1 \right\}\)	\((28,\ 30]\)
\(0\)	\(\left\{ +31,\ -1 \right\}\)	\((30,\ \infty)\)
\(1\)	\(\left\{ +1,\ +17 \right\}\)	\((0,\ 2]\)
\(1\)	\(\left\{ +3,\ +17 \right\}\)	\((2,\ 4]\)
\(1\)	\(\left\{ +5,\ +17 \right\}\)	\((4,\ 6]\)
\(1\)	\(\left\{ +7,\ +17 \right\}\)	\((6,\ 8]\)
\(1\)	\(\left\{ +9,\ +17 \right\}\)	\((8,\ 10]\)
\(1\)	\(\left\{ +11,\ +17 \right\}\)	\((10,\ 12]\)
\(1\)	\(\left\{ +13,\ +17 \right\}\)	\((12,\ 14]\)
\(1\)	\(\left\{ +15,\ +17 \right\}\)	\((14,\ 18]\)
\(1\)	\(\left\{ +15,\ +19 \right\}\)	\((18,\ 20]\)
\(1\)	\(\left\{ +15,\ +21 \right\}\)	\((20,\ 22]\)
\(1\)	\(\left\{ +15,\ +23 \right\}\)	\((22,\ 24]\)
\(1\)	\(\left\{ +15,\ +25 \right\}\)	\((24,\ 26]\)
\(1\)	\(\left\{ +15,\ +27 \right\}\)	\((26,\ 28]\)
\(1\)	\(\left\{ +15,\ +29 \right\}\)	\((28,\ 30]\)
\(1\)	\(\left\{ +15,\ +31 \right\}\)	\((30,\ \infty)\)
\(2\)	\(\left\{ +9,\ +1 \right\}\)	\((0,\ 2]\)
\(2\)	\(\left\{ +9,\ +3 \right\}\)	\((2,\ 4]\)
\(2\)	\(\left\{ +9,\ +5 \right\}\)	\((4,\ 6]\)
\(2\)	\(\left\{ +9,\ +7 \right\}\)	\((6,\ 10]\)
\(2\)	\(\left\{ +11,\ +7 \right\}\)	\((10,\ 12]\)
\(2\)	\(\left\{ +13,\ +7 \right\}\)	\((12,\ 14]\)
\(2\)	\(\left\{ +15,\ +7 \right\}\)	\((14,\ 16]\)
\(2\)	\(\left\{ +17,\ +25 \right\}\)	\((16,\ 18]\)
\(2\)	\(\left\{ +19,\ +25 \right\}\)	\((18,\ 20]\)
\(2\)	\(\left\{ +21,\ +25 \right\}\)	\((20,\ 22]\)
\(2\)	\(\left\{ +23,\ +25 \right\}\)	\((22,\ 26]\)
\(2\)	\(\left\{ +23,\ +27 \right\}\)	\((26,\ 28]\)
\(2\)	\(\left\{ +23,\ +29 \right\}\)	\((28,\ 30]\)
\(2\)	\(\left\{ +23,\ +31 \right\}\)	\((30,\ \infty)\)
\(3\)	\(\left\{ +5,\ +1 \right\}\)	\((0,\ 2]\)
\(3\)	\(\left\{ +5,\ +3 \right\}\)	\((2,\ 6]\)
\(3\)	\(\left\{ +7,\ +3 \right\}\)	\((6,\ 8]\)
\(3\)	\(\left\{ +9,\ +13 \right\}\)	\((8,\ 10]\)
\(3\)	\(\left\{ +11,\ +13 \right\}\)	\((10,\ 14]\)
\(3\)	\(\left\{ +11,\ +15 \right\}\)	\((14,\ 16]\)
\(3\)	\(\left\{ +21,\ +17 \right\}\)	\((16,\ 18]\)
\(3\)	\(\left\{ +21,\ +19 \right\}\)	\((18,\ 22]\)
\(3\)	\(\left\{ +23,\ +19 \right\}\)	\((22,\ 24]\)
\(3\)	\(\left\{ +25,\ +29 \right\}\)	\((24,\ 26]\)
\(3\)	\(\left\{ +27,\ +29 \right\}\)	\((26,\ 30]\)
\(3\)	\(\left\{ +27,\ +31 \right\}\)	\((30,\ \infty)\)
\(4\)	\(\left\{ +3,\ +1 \right\}\)	\((0,\ 4]\)
\(4\)	\(\left\{ +5,\ +7 \right\}\)	\((4,\ 8]\)
\(4\)	\(\left\{ +11,\ +9 \right\}\)	\((8,\ 12]\)
\(4\)	\(\left\{ +13,\ +15 \right\}\)	\((12,\ 16]\)
\(4\)	\(\left\{ +19,\ +17 \right\}\)	\((16,\ 20]\)
\(4\)	\(\left\{ +21,\ +23 \right\}\)	\((20,\ 24]\)
\(4\)	\(\left\{ +27,\ +25 \right\}\)	\((24,\ 28]\)
\(4\)	\(\left\{ +29,\ +31 \right\}\)	\((28,\ \infty)\)

A.3 4096QAM

\(i_b\)	\(x_{\mathrm{I},\min}\)	\(\hat{x}_{\mathrm{I}}\)
\(0\)	\(\left\{ +1,\ -1 \right\}\)	\((0,\ 2]\)
\(0\)	\(\left\{ +3,\ -1 \right\}\)	\((2,\ 4]\)
\(0\)	\(\left\{ +5,\ -1 \right\}\)	\((4,\ 6]\)
\(0\)	\(\left\{ +7,\ -1 \right\}\)	\((6,\ 8]\)
\(0\)	\(\left\{ +9,\ -1 \right\}\)	\((8,\ 10]\)
\(0\)	\(\left\{ +11,\ -1 \right\}\)	\((10,\ 12]\)
\(0\)	\(\left\{ +13,\ -1 \right\}\)	\((12,\ 14]\)
\(0\)	\(\left\{ +15,\ -1 \right\}\)	\((14,\ 16]\)
\(0\)	\(\left\{ +17,\ -1 \right\}\)	\((16,\ 18]\)
\(0\)	\(\left\{ +19,\ -1 \right\}\)	\((18,\ 20]\)
\(0\)	\(\left\{ +21,\ -1 \right\}\)	\((20,\ 22]\)
\(0\)	\(\left\{ +23,\ -1 \right\}\)	\((22,\ 24]\)
\(0\)	\(\left\{ +25,\ -1 \right\}\)	\((24,\ 26]\)
\(0\)	\(\left\{ +27,\ -1 \right\}\)	\((26,\ 28]\)
\(0\)	\(\left\{ +29,\ -1 \right\}\)	\((28,\ 30]\)
\(0\)	\(\left\{ +31,\ -1 \right\}\)	\((30,\ 32]\)
\(0\)	\(\left\{ +33,\ -1 \right\}\)	\((32,\ 34]\)
\(0\)	\(\left\{ +35,\ -1 \right\}\)	\((34,\ 36]\)
\(0\)	\(\left\{ +37,\ -1 \right\}\)	\((36,\ 38]\)
\(0\)	\(\left\{ +39,\ -1 \right\}\)	\((38,\ 40]\)
\(0\)	\(\left\{ +41,\ -1 \right\}\)	\((40,\ 42]\)
\(0\)	\(\left\{ +43,\ -1 \right\}\)	\((42,\ 44]\)
\(0\)	\(\left\{ +45,\ -1 \right\}\)	\((44,\ 46]\)
\(0\)	\(\left\{ +47,\ -1 \right\}\)	\((46,\ 48]\)
\(0\)	\(\left\{ +49,\ -1 \right\}\)	\((48,\ 50]\)
\(0\)	\(\left\{ +51,\ -1 \right\}\)	\((50,\ 52]\)
\(0\)	\(\left\{ +53,\ -1 \right\}\)	\((52,\ 54]\)
\(0\)	\(\left\{ +55,\ -1 \right\}\)	\((54,\ 56]\)
\(0\)	\(\left\{ +57,\ -1 \right\}\)	\((56,\ 58]\)
\(0\)	\(\left\{ +59,\ -1 \right\}\)	\((58,\ 60]\)
\(0\)	\(\left\{ +61,\ -1 \right\}\)	\((60,\ 62]\)
\(0\)	\(\left\{ +63,\ -1 \right\}\)	\((62,\ \infty)\)
\(1\)	\(\left\{ +1,\ +33 \right\}\)	\((0,\ 2]\)
\(1\)	\(\left\{ +3,\ +33 \right\}\)	\((2,\ 4]\)
\(1\)	\(\left\{ +5,\ +33 \right\}\)	\((4,\ 6]\)
\(1\)	\(\left\{ +7,\ +33 \right\}\)	\((6,\ 8]\)
\(1\)	\(\left\{ +9,\ +33 \right\}\)	\((8,\ 10]\)
\(1\)	\(\left\{ +11,\ +33 \right\}\)	\((10,\ 12]\)
\(1\)	\(\left\{ +13,\ +33 \right\}\)	\((12,\ 14]\)
\(1\)	\(\left\{ +15,\ +33 \right\}\)	\((14,\ 16]\)
\(1\)	\(\left\{ +17,\ +33 \right\}\)	\((16,\ 18]\)
\(1\)	\(\left\{ +19,\ +33 \right\}\)	\((18,\ 20]\)
\(1\)	\(\left\{ +21,\ +33 \right\}\)	\((20,\ 22]\)
\(1\)	\(\left\{ +23,\ +33 \right\}\)	\((22,\ 24]\)
\(1\)	\(\left\{ +25,\ +33 \right\}\)	\((24,\ 26]\)
\(1\)	\(\left\{ +27,\ +33 \right\}\)	\((26,\ 28]\)
\(1\)	\(\left\{ +29,\ +33 \right\}\)	\((28,\ 30]\)
\(1\)	\(\left\{ +31,\ +33 \right\}\)	\((30,\ 34]\)
\(1\)	\(\left\{ +31,\ +35 \right\}\)	\((34,\ 36]\)
\(1\)	\(\left\{ +31,\ +37 \right\}\)	\((36,\ 38]\)
\(1\)	\(\left\{ +31,\ +39 \right\}\)	\((38,\ 40]\)
\(1\)	\(\left\{ +31,\ +41 \right\}\)	\((40,\ 42]\)
\(1\)	\(\left\{ +31,\ +43 \right\}\)	\((42,\ 44]\)
\(1\)	\(\left\{ +31,\ +45 \right\}\)	\((44,\ 46]\)
\(1\)	\(\left\{ +31,\ +47 \right\}\)	\((46,\ 48]\)
\(1\)	\(\left\{ +31,\ +49 \right\}\)	\((48,\ 50]\)
\(1\)	\(\left\{ +31,\ +51 \right\}\)	\((50,\ 52]\)
\(1\)	\(\left\{ +31,\ +53 \right\}\)	\((52,\ 54]\)
\(1\)	\(\left\{ +31,\ +55 \right\}\)	\((54,\ 56]\)
\(1\)	\(\left\{ +31,\ +57 \right\}\)	\((56,\ 58]\)
\(1\)	\(\left\{ +31,\ +59 \right\}\)	\((58,\ 60]\)
\(1\)	\(\left\{ +31,\ +61 \right\}\)	\((60,\ 62]\)
\(1\)	\(\left\{ +31,\ +63 \right\}\)	\((62,\ \infty)\)
\(2\)	\(\left\{ +17,\ +1 \right\}\)	\((0,\ 2]\)
\(2\)	\(\left\{ +17,\ +3 \right\}\)	\((2,\ 4]\)
\(2\)	\(\left\{ +17,\ +5 \right\}\)	\((4,\ 6]\)
\(2\)	\(\left\{ +17,\ +7 \right\}\)	\((6,\ 8]\)
\(2\)	\(\left\{ +17,\ +9 \right\}\)	\((8,\ 10]\)
\(2\)	\(\left\{ +17,\ +11 \right\}\)	\((10,\ 12]\)
\(2\)	\(\left\{ +17,\ +13 \right\}\)	\((12,\ 14]\)
\(2\)	\(\left\{ +17,\ +15 \right\}\)	\((14,\ 18]\)
\(2\)	\(\left\{ +19,\ +15 \right\}\)	\((18,\ 20]\)
\(2\)	\(\left\{ +21,\ +15 \right\}\)	\((20,\ 22]\)
\(2\)	\(\left\{ +23,\ +15 \right\}\)	\((22,\ 24]\)
\(2\)	\(\left\{ +25,\ +15 \right\}\)	\((24,\ 26]\)
\(2\)	\(\left\{ +27,\ +15 \right\}\)	\((26,\ 28]\)
\(2\)	\(\left\{ +29,\ +15 \right\}\)	\((28,\ 30]\)
\(2\)	\(\left\{ +31,\ +15 \right\}\)	\((30,\ 32]\)
\(2\)	\(\left\{ +33,\ +49 \right\}\)	\((32,\ 34]\)
\(2\)	\(\left\{ +35,\ +49 \right\}\)	\((34,\ 36]\)
\(2\)	\(\left\{ +37,\ +49 \right\}\)	\((36,\ 38]\)
\(2\)	\(\left\{ +39,\ +49 \right\}\)	\((38,\ 40]\)
\(2\)	\(\left\{ +41,\ +49 \right\}\)	\((40,\ 42]\)
\(2\)	\(\left\{ +43,\ +49 \right\}\)	\((42,\ 44]\)
\(2\)	\(\left\{ +45,\ +49 \right\}\)	\((44,\ 46]\)
\(2\)	\(\left\{ +47,\ +49 \right\}\)	\((46,\ 50]\)
\(2\)	\(\left\{ +47,\ +51 \right\}\)	\((50,\ 52]\)
\(2\)	\(\left\{ +47,\ +53 \right\}\)	\((52,\ 54]\)
\(2\)	\(\left\{ +47,\ +55 \right\}\)	\((54,\ 56]\)
\(2\)	\(\left\{ +47,\ +57 \right\}\)	\((56,\ 58]\)
\(2\)	\(\left\{ +47,\ +59 \right\}\)	\((58,\ 60]\)
\(2\)	\(\left\{ +47,\ +61 \right\}\)	\((60,\ 62]\)
\(2\)	\(\left\{ +47,\ +63 \right\}\)	\((62,\ \infty)\)
\(3\)	\(\left\{ +9,\ +1 \right\}\)	\((0,\ 2]\)
\(3\)	\(\left\{ +9,\ +3 \right\}\)	\((2,\ 4]\)
\(3\)	\(\left\{ +9,\ +5 \right\}\)	\((4,\ 6]\)
\(3\)	\(\left\{ +9,\ +7 \right\}\)	\((6,\ 10]\)
\(3\)	\(\left\{ +11,\ +7 \right\}\)	\((10,\ 12]\)
\(3\)	\(\left\{ +13,\ +7 \right\}\)	\((12,\ 14]\)
\(3\)	\(\left\{ +15,\ +7 \right\}\)	\((14,\ 16]\)
\(3\)	\(\left\{ +17,\ +25 \right\}\)	\((16,\ 18]\)
\(3\)	\(\left\{ +19,\ +25 \right\}\)	\((18,\ 20]\)
\(3\)	\(\left\{ +21,\ +25 \right\}\)	\((20,\ 22]\)
\(3\)	\(\left\{ +23,\ +25 \right\}\)	\((22,\ 26]\)
\(3\)	\(\left\{ +23,\ +27 \right\}\)	\((26,\ 28]\)
\(3\)	\(\left\{ +23,\ +29 \right\}\)	\((28,\ 30]\)
\(3\)	\(\left\{ +23,\ +31 \right\}\)	\((30,\ 32]\)
\(3\)	\(\left\{ +41,\ +33 \right\}\)	\((32,\ 34]\)
\(3\)	\(\left\{ +41,\ +35 \right\}\)	\((34,\ 36]\)
\(3\)	\(\left\{ +41,\ +37 \right\}\)	\((36,\ 38]\)
\(3\)	\(\left\{ +41,\ +39 \right\}\)	\((38,\ 42]\)
\(3\)	\(\left\{ +43,\ +39 \right\}\)	\((42,\ 44]\)
\(3\)	\(\left\{ +45,\ +39 \right\}\)	\((44,\ 46]\)
\(3\)	\(\left\{ +47,\ +39 \right\}\)	\((46,\ 48]\)
\(3\)	\(\left\{ +49,\ +57 \right\}\)	\((48,\ 50]\)
\(3\)	\(\left\{ +51,\ +57 \right\}\)	\((50,\ 52]\)
\(3\)	\(\left\{ +53,\ +57 \right\}\)	\((52,\ 54]\)
\(3\)	\(\left\{ +55,\ +57 \right\}\)	\((54,\ 58]\)
\(3\)	\(\left\{ +55,\ +59 \right\}\)	\((58,\ 60]\)
\(3\)	\(\left\{ +55,\ +61 \right\}\)	\((60,\ 62]\)
\(3\)	\(\left\{ +55,\ +63 \right\}\)	\((62,\ \infty)\)
\(4\)	\(\left\{ +5,\ +1 \right\}\)	\((0,\ 2]\)
\(4\)	\(\left\{ +5,\ +3 \right\}\)	\((2,\ 6]\)
\(4\)	\(\left\{ +7,\ +3 \right\}\)	\((6,\ 8]\)
\(4\)	\(\left\{ +9,\ +13 \right\}\)	\((8,\ 10]\)
\(4\)	\(\left\{ +11,\ +13 \right\}\)	\((10,\ 14]\)
\(4\)	\(\left\{ +11,\ +15 \right\}\)	\((14,\ 16]\)
\(4\)	\(\left\{ +21,\ +17 \right\}\)	\((16,\ 18]\)
\(4\)	\(\left\{ +21,\ +19 \right\}\)	\((18,\ 22]\)
\(4\)	\(\left\{ +23,\ +19 \right\}\)	\((22,\ 24]\)
\(4\)	\(\left\{ +25,\ +29 \right\}\)	\((24,\ 26]\)
\(4\)	\(\left\{ +27,\ +29 \right\}\)	\((26,\ 30]\)
\(4\)	\(\left\{ +27,\ +31 \right\}\)	\((30,\ 32]\)
\(4\)	\(\left\{ +37,\ +33 \right\}\)	\((32,\ 34]\)
\(4\)	\(\left\{ +37,\ +35 \right\}\)	\((34,\ 38]\)
\(4\)	\(\left\{ +39,\ +35 \right\}\)	\((38,\ 40]\)
\(4\)	\(\left\{ +41,\ +45 \right\}\)	\((40,\ 42]\)
\(4\)	\(\left\{ +43,\ +45 \right\}\)	\((42,\ 46]\)
\(4\)	\(\left\{ +43,\ +47 \right\}\)	\((46,\ 48]\)
\(4\)	\(\left\{ +53,\ +49 \right\}\)	\((48,\ 50]\)
\(4\)	\(\left\{ +53,\ +51 \right\}\)	\((50,\ 54]\)
\(4\)	\(\left\{ +55,\ +51 \right\}\)	\((54,\ 56]\)
\(4\)	\(\left\{ +57,\ +61 \right\}\)	\((56,\ 58]\)
\(4\)	\(\left\{ +59,\ +61 \right\}\)	\((58,\ 62]\)
\(4\)	\(\left\{ +59,\ +63 \right\}\)	\((62,\ \infty)\)
\(5\)	\(\left\{ +3,\ +1 \right\}\)	\((0,\ 4]\)
\(5\)	\(\left\{ +5,\ +7 \right\}\)	\((4,\ 8]\)
\(5\)	\(\left\{ +11,\ +9 \right\}\)	\((8,\ 12]\)
\(5\)	\(\left\{ +13,\ +15 \right\}\)	\((12,\ 16]\)
\(5\)	\(\left\{ +19,\ +17 \right\}\)	\((16,\ 20]\)
\(5\)	\(\left\{ +21,\ +23 \right\}\)	\((20,\ 24]\)
\(5\)	\(\left\{ +27,\ +25 \right\}\)	\((24,\ 28]\)
\(5\)	\(\left\{ +29,\ +31 \right\}\)	\((28,\ 32]\)
\(5\)	\(\left\{ +35,\ +33 \right\}\)	\((32,\ 36]\)
\(5\)	\(\left\{ +37,\ +39 \right\}\)	\((36,\ 40]\)
\(5\)	\(\left\{ +43,\ +41 \right\}\)	\((40,\ 44]\)
\(5\)	\(\left\{ +45,\ +47 \right\}\)	\((44,\ 48]\)
\(5\)	\(\left\{ +51,\ +49 \right\}\)	\((48,\ 52]\)
\(5\)	\(\left\{ +53,\ +55 \right\}\)	\((52,\ 56]\)
\(5\)	\(\left\{ +59,\ +57 \right\}\)	\((56,\ 60]\)
\(5\)	\(\left\{ +61,\ +63 \right\}\)	\((60,\ \infty)\)