===== Main Spectral Data Encoding Trees ===== ==== General Intro ==== The quantized values are encoded using a prefix free binary code. To make matters more complicated there are up to 16 encoding scheme per quantization precision (symbol set depends on precision, so different encoding schemes must be used for different precisions.). In many cases, the symbols are built from a group of two or four quantized values instead of just one quantized value. For groups of size two, the trees are shown as table, the first symbol choosing the row, the second value the column. For groups of size four, the tables are organized so that the first symbol chooses the row block, the second symbol the exact row, the third symbol the column block and the fourth symbol the column. //I think about swapping second and third symbol, so be careful when coming back later!// Another catch is that some of the encoding schemes encoded the signed quantized values, while others encode only the absolute values of the quantized values. If only the absolute values are encoded in a symbol, each symbol is followed by one bit per absolute value that is not zero. If that bit is set, the value is negative. The trees can be calculated as [[canonical huffman trees]]. Note that unsigned sorting (on two's complement numbers) is used, so negative numbers sort behind positive ones. The sort order for the grouped symbols is lexicographically by the single quantized numbers. ==== Examples ==== === Interpreting a group-by-four block === Let's take a look at Tree 1A. The shortest symbol is (0,0,0,0), encoded with one bit, and gets assigned the code ''0''. The next shortest symbol has 4 bits, there is only one of them, and it is (-1,0,0,0). So this symbol gets the code ''1000''. Now, there are seven symbols with 5 bits. From top to bottom in the table, the symbols are * (0,-1,0,0) * (0,0,-1,0) * (0,0,0,-1) * (0,0,0,1) * (0,0,1,0) * (0,1,0,0) * (1,0,0,0) Now for sorting these symbols lexicographically, rember that negative values go past positive ones (if we had -2,-1,0,1,2 the sorted order would be 0,1,2,-2,-1 instead). In this case, the sorted order ist * (0,0,0,1) -> ''10010'' * (0,0,0,-1) -> ''10011'' * (0,0,1,0) -> ''10100'' * (0,0,-1,0) -> ''10101'' * (0,1,0,0) -> ''10110'' * (0,-1,0,0) -> ''10111'' * (1,0,0,0) -> ''11000'' === Interpreting absolute value encoding === Now take a look at Tree 1H. In this coding scheme, the prefix free code is built from positive values only. Applying the algorithm to build the canonical code from the given lengths, one gets the following code assignment: * (0,0) -> ''0'' * (1,0) -> ''10'' * (0,1) -> ''110'' * (1,1) -> ''111'' as the these codes are absolute value, they have to be interpreted as (0,0), (±1,0), (0,±1), (±1,±1) in fact, and the signs have to be specified afterwards. For each non-zero value from left to right, one bit being zero for positive and one for negative is appended to to the code, so you finally obtain * (0,0) -> ''0'' * (+1,0) -> ''10**0**''; (-1,0) -> ''10**1**'' * (0,+1) -> ''110**0**''; (0,-1) -> ''110**1**'' * (+1,+1) -> ''111**00**'', (-1,+1) -> ''111**10**'', (+1,-1) -> ''111**01**'', (-1,-1) -> ''111**11**'' ==== Trees for Precision 1 ==== (i.e. symbol set -1..+1) === Tree 1A === (used for A/0)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	11	9	12	10	7	10	12	10	11
	0	10	8	10	7	4	8	9	7	10
	1	12	9	10	9	7	10	11	10	12
0	-1	9	7	10	8	5	7	9	7	10
	0	7	5	7	5	1	5	7	5	7
	1	10	7	9	7	5	8	10	7	9
1	-1	12	10	11	9	7	9	10	9	12
	0	10	7	9	7	5	7	10	8	10
	1	11	10	12	10	7	10	12	9	11

=== Tree 1B === (used for A/1)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	10	9	11	10	7	10	11	10	11
	0	10	8	9	7	5	7	9	7	10
	1	11	8	9	9	7	9	11	9	11
0	-1	9	8	9	7	5	7	8	7	9
	0	7	5	7	5	1	5	6	5	7
	1	9	7	8	7	5	8	9	8	9
1	-1	11	9	11	9	7	9	9	8	11
	0	9	7	9	7	5	7	9	8	10
	1	11	9	11	10	7	10	11	9	10

=== Tree 1C === (used for A/2, B/1)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	10	9	12	10	8	10	11	10	11
	0	10	8	9	8	5	8	9	7	10
	1	11	8	9	9	7	9	11	10	12
0	-1	9	8	10	8	5	8	8	7	10
	0	8	5	7	4	1	4	7	5	8
	1	9	7	8	8	5	8	10	8	9
1	-1	12	10	11	9	7	9	9	8	11
	0	10	8	9	8	5	8	9	8	10
	1	11	10	11	10	8	10	12	9	10

=== Tree 1D === (used for A/3)

	-1	0	1
-1	6	4	6
0	3	1	3
1	6	3	6

=== Tree 1E === (used for A/4, B/2, B/6)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	12	11	12	12	9	12	12	11	12
	0	12	10	11	9	5	9	11	9	12
	1	12	10	11	11	9	11	12	11	12
0	-1	11	10	11	9	4	9	10	9	11
	0	9	4	9	4	1	4	9	4	9
	1	11	9	10	9	4	10	11	10	11
1	-1	12	11	12	11	9	11	11	10	12
	0	11	9	11	9	5	9	11	10	12
	1	12	11	12	11	9	12	12	11	12

=== Tree 1F === (used for A/5, B/3)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	10	9	11	10	7	9	11	9	11
	0	10	7	9	7	4	7	9	7	9
	1	11	8	9	9	6	9	11	9	11
0	-1	9	7	9	7	5	7	8	6	9
	0	7	4	6	4	2	4	6	5	7
	1	9	6	8	7	4	7	9	7	9
1	-1	11	9	11	9	6	9	9	8	11
	0	9	7	9	7	4	7	9	7	10
	1	11	9	11	9	7	10	11	9	10

=== Tree 1G === (used for A/6, B/5)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	10	9	11	9	7	9	11	9	11
	0	9	8	9	8	4	8	9	7	9
	1	10	9	10	9	7	9	10	9	10
0	-1	8	7	9	8	4	7	8	7	9
	0	7	4	6	4	2	4	7	4	7
	1	9	7	8	7	4	7	9	7	9
1	-1	10	9	11	9	7	9	10	9	10
	0	9	7	9	8	4	8	9	8	9
	1	10	9	10	9	7	9	10	9	10

=== Tree 1H === (used for A/7) (absolute values in followed by sign bits if nonzero)

	0	1
0	1	3
1	2	3

=== Tree 1I === (used for B/0)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	10	9	12	10	7	10	12	10	11
	0	10	8	10	8	4	8	9	7	10
	1	12	9	10	9	7	10	11	10	12
0	-1	9	8	10	8	5	8	9	7	10
	0	7	5	7	5	1	5	7	5	7
	1	10	7	9	8	5	8	10	8	9
1	-1	12	10	11	10	7	9	10	9	12
	0	10	8	9	8	4	8	10	8	10
	1	11	10	12	10	7	10	12	9	10

=== Tree 1J === (used for B/4)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	11	9	12	10	7	10	12	10	12
	0	10	7	10	7	5	8	10	7	10
	1	12	9	10	10	7	10	12	10	12
0	-1	9	7	10	8	5	7	9	7	10
	0	7	5	7	5	1	5	7	5	7
	1	10	7	9	7	5	8	10	7	9
1	-1	12	10	12	10	7	9	10	9	12
	0	10	7	9	7	4	8	10	7	10
	1	12	10	12	10	7	10	12	9	11

=== Tree 1K === (used for B/7)

		-1			0			1
		-1	0	1	-1	0	1	-1	0	1
-1	-1	10	9	12	10	7	9	12	10	12
	0	10	7	10	8	4	8	10	7	9
	1	12	9	10	9	7	10	12	10	12
0	-1	9	7	10	8	5	7	9	7	10
	0	7	5	7	5	1	5	7	5	7
	1	10	7	9	7	5	8	10	7	9
1	-1	12	10	12	10	7	10	10	9	12
	0	9	7	10	7	5	8	10	7	10
	1	12	10	12	9	7	10	12	9	10

==== Trees for Precision 2 ==== (i.e. symbol set -2..+2) === Tree 2A === (used for A/0) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	2	4	8	4	5	7	8	8	10
	1	4	5	8	5	5	8	8	8	10
	2	8	8	9	8	8	10	10	10	11
1	0	4	5	8	5	5	8	8	9	10
	1	5	5	8	5	5	8	9	8	10
	2	8	8	10	8	8	10	9	9	10
2	0	8	8	10	8	8	10	10	10	12
	1	8	8	11	8	8	10	10	10	11
	2	10	10	11	10	9	11	11	11	12

=== Tree 2B === (used for A/1) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	5	5	7	5	5	6	8	7	8
	1	5	5	7	5	5	6	7	7	7
	2	7	7	8	7	6	8	8	7	8
1	0	5	5	7	5	5	6	7	7	7
	1	5	5	7	5	5	6	7	7	7
	2	7	7	8	6	6	7	8	7	7
2	0	7	7	8	7	7	8	8	8	9
	1	6	6	8	6	6	7	7	7	8
	2	8	8	10	7	7	8	10	8	8

=== Tree 2C === (used for A/2)

	-2	-1	0	1	2
-2	9	8	6	7	8
-1	7	5	3	5	7
0	7	3	2	3	6
1	7	5	3	5	7
2	8	7	6	7	9

=== Tree 2D === (used for A/3, B/2) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	1	4	6	5	6	7	6	7	8
	1	5	7	8	6	7	8	7	8	8
	2	7	8	10	7	8	10	8	9	9
1	0	4	7	9	7	8	9	8	9	9
	1	6	8	9	7	8	9	8	9	9
	2	7	9	10	8	9	10	9	9	10
2	0	6	9	10	8	9	10	10	10	11
	1	7	9	10	8	9	11	10	10	11
	2	8	10	11	8	10	11	9	10	11

=== Tree 2E === (used for A/4) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	3	4	7	5	5	7	7	7	8
	1	5	5	7	5	5	7	7	7	8
	2	7	7	9	7	7	9	9	8	10
1	0	4	5	7	5	5	7	7	7	8
	1	5	5	7	5	5	7	7	7	8
	2	7	7	9	7	6	8	8	8	9
2	0	7	7	9	7	7	9	9	9	10
	1	7	7	9	7	7	8	9	8	9
	2	9	8	10	8	8	9	9	9	10

=== Tree 2F === (used for A/5)

	-2	-1	0	1	2
-2	10	9	7	8	9
-1	8	6	4	6	8
0	8	3	1	3	7
1	8	6	4	6	8
2	9	8	7	8	10

=== Tree 2G === (used for A/6)

	-2	-1	0	1	2
-2	8	7	6	6	7
-1	6	5	3	4	6
0	6	3	3	3	6
1	6	4	3	5	7
2	7	6	6	7	8

=== Tree 2H === (used for A/7) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	2	4	7	4	5	7	7	8	9
	1	4	5	8	5	5	8	8	7	9
	2	7	8	10	8	8	9	9	9	10
1	0	4	5	8	5	6	8	8	8	9
	1	5	6	8	5	6	8	8	8	9
	2	8	8	10	7	8	9	9	9	10
2	0	7	8	10	8	8	10	10	10	11
	1	7	8	10	8	8	9	9	9	10
	2	9	9	11	9	9	11	10	10	11

=== Tree 2I === (used for B/0) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	1	4	7	4	6	8	8	8	10
	1	4	6	9	6	6	9	8	9	10
	2	8	9	11	8	9	11	10	10	12
1	0	4	6	9	6	7	9	9	9	11
	1	6	7	10	6	7	9	9	9	10
	2	8	10	11	9	9	11	10	10	11
2	0	8	9	12	9	10	12	11	11	12
	1	8	9	12	9	9	11	11	11	12
	2	10	11	12	10	10	12	12	11	12

=== Tree 2J === (used for B/1) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	3	4	7	5	5	7	7	7	9
	1	4	5	7	5	5	7	7	7	8
	2	7	7	9	7	8	9	9	8	10
1	0	4	5	7	5	5	7	8	7	9
	1	5	5	7	5	5	7	7	7	8
	2	7	7	9	7	7	9	8	8	9
2	0	7	7	10	7	7	9	9	9	10
	1	7	7	9	7	7	9	9	9	10
	2	9	9	10	8	8	10	10	9	10

=== Tree 2K === (used for B/3, B/7) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	2	4	6	4	5	7	7	7	8
	1	4	5	7	5	6	8	7	8	9
	2	7	8	10	7	8	10	9	9	10
1	0	4	5	8	5	6	8	8	8	9
	1	5	6	8	6	6	8	8	8	9
	2	7	8	10	8	8	10	9	9	10
2	0	6	8	10	7	8	10	10	10	11
	1	7	8	10	8	8	10	9	10	11
	2	8	9	11	9	9	10	10	10	11

=== Tree 2L === (used for B/4)

	-2	-1	0	1	2
-2	9	8	7	7	9
-1	8	6	3	5	7
0	7	4	1	4	6
1	7	5	4	6	8
2	9	7	7	8	9

=== Tree 2M === (used for B/5)

	-2	-1	0	1	2
-2	9	8	7	7	9
-1	8	6	4	5	7
0	6	4	1	4	7
1	7	5	3	6	8
2	9	7	7	8	9

=== Tree 2N === (used for B/6) (absolute values in followed by sign bits if nonzero)

		0			1			2
		0	1	2	0	1	2	0	1	2
0	0	1	4	6	5	6	6	7	7	8
	1	5	7	8	7	7	8	7	8	8
	2	7	9	10	7	9	10	8	9	10
1	0	4	7	9	7	8	9	9	9	10
	1	6	8	9	7	8	9	9	9	10
	2	7	9	11	8	9	10	9	9	10
2	0	5	9	11	8	10	11	10	11	12
	1	6	9	11	8	9	11	10	10	11
	2	8	10	11	8	10	12	10	10	11

==== Trees for Precision 3 ==== (i.e. symbol set -3..+3) === Tree 3A === (used for A/0, B/2)

	-3	-2	-1	0	1	2	3
-3	9	8	7	6	7	7	8
-2	8	7	6	6	6	7	7
-1	7	6	4	4	4	6	7
0	7	6	4	3	4	6	7
1	7	6	4	4	4	6	7
2	7	7	6	6	6	8	8
3	8	7	7	7	7	8	9

=== Tree 3B === (used for A/1, B/1)

	-3	-2	-1	0	1	2	3
-3	10	9	8	8	8	9	10
-2	9	9	7	7	7	8	9
-1	9	8	6	4	5	7	8
0	8	7	4	1	4	7	8
1	8	7	5	4	6	8	9
2	9	8	7	7	8	9	9
3	10	9	8	8	8	9	10

=== Tree 3C === (used for A/2)

	-3	-2	-1	0	1	2	3
-3	9	9	8	7	8	8	9
-2	9	8	7	6	6	8	8
-1	8	7	5	3	5	7	8
0	7	6	3	3	3	6	7
1	8	7	5	3	5	7	8
2	8	8	6	6	7	8	9
3	9	8	8	7	8	9	9

=== Tree 3D === (used for A/3) (absolute values in followed by sign bits if nonzero)

		0				1				2				3
		0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3
0	0	3	5	8	10	4	5	7	9	7	8	10	10	12	10	10	12
	1	5	5	8	9	5	5	7	8	8	7	10	10	10	9	10	9
	2	8	8	9	9	7	8	10	10	9	8	10	11	10	10	11	11
	3	10	10	11	12	10	10	10	12	11	10	12	12	11	10	12	11
1	0	5	5	8	10	5	5	8	10	7	8	9	10	10	9	11	12
	1	5	5	8	9	5	6	7	9	8	8	8	10	10	9	10	10
	2	8	8	10	11	7	8	9	11	8	7	10	11	11	10	9	10
	3	10	9	12	10	9	11	10	10	9	9	10	11	10	10	11	12
2	0	7	8	10	10	8	7	10	10	9	9	11	10	11	10	12	10
	1	7	7	10	12	7	8	8	10	11	9	8	10	11	10	10	12
	2	8	9	10	12	9	10	9	10	11	9	11	10	12	12	11	10
	3	11	10	9	12	10	9	10	12	11	11	12	10	12	12	12	12
3	0	10	9	11	11	10	10	11	12	9	11	12	12	12	12	11	12
	1	9	9	12	12	8	9	10	12	12	12	11	12	11	12	10	12
	2	10	10	10	12	8	10	10	12	12	9	12	10	12	10	12	12
	3	10	10	12	12	10	10	10	12	10	12	12	12	10	12	10	12

=== Tree 3E === (used for A/4) * -3: 4 bits * -2: 3 bits * -1,0: 2 bits * 1,2: 3 bits * 3: 4 bits === Tree 3F === (used for A/5) * -3: 4 bits * -2,-1: 3 bits * 0,1: 2 bits * 2: 3 bits * 3: 4 bits === Tree 3G === (used for A/6) (absolute values in followed by sign bits if nonzero)

		0				1				2				3
		0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3
0	0	3	4	7	9	5	5	7	10	8	7	9	9	10	10	11	11
	1	5	5	8	9	5	6	8	9	7	7	7	10	10	9	11	10
	2	9	7	11	10	8	8	10	11	10	10	10	10	10	9	10	10
	3	9	9	11	11	9	9	10	11	10	10	11	11	11	10	11	11
1	0	5	5	8	11	6	6	8	9	9	7	10	9	11	9	10	11
	1	5	5	8	9	6	5	7	10	7	8	7	9	10	9	9	11
	2	8	8	9	10	8	8	8	10	8	8	10	10	10	8	10	10
	3	11	11	11	10	10	8	11	11	10	11	10	11	11	10	9	11
2	0	7	7	11	10	8	7	9	10	9	8	10	11	11	11	10	10
	1	8	8	8	10	7	7	9	11	10	8	9	11	9	10	10	11
	2	9	9	9	11	8	8	9	10	9	8	10	10	10	9	11	10
	3	10	10	11	11	11	11	11	9	10	9	11	11	11	10	10	10
3	0	10	9	10	11	9	9	11	11	9	10	10	11	11	11	10	11
	1	9	8	10	11	8	10	10	10	10	11	11	11	11	9	11	11
	2	10	9	10	11	9	10	9	11	10	9	10	11	11	11	11	11
	3	11	11	11	10	10	11	10	11	9	11	11	11	11	10	10	11

=== Tree 3H === (used for A/7)

	-3	-2	-1	0	1	2	3
-3	10	9	9	8	7	8	10
-2	9	9	7	6	6	7	8
-1	8	7	5	4	4	6	8
0	8	6	4	2	3	6	8
1	8	6	4	4	5	7	8
2	8	7	6	6	7	8	9
3	9	9	8	8	9	10	10

=== Tree 3I === (used for B/0)

	-3	-2	-1	0	1	2	3
-3	9	9	8	7	8	8	9
-2	9	9	8	7	7	9	9
-1	8	7	6	4	6	7	8
0	7	6	4	1	4	7	7
1	8	7	6	4	6	8	8
2	9	8	7	7	8	9	9
3	9	9	8	7	8	9	9

=== Tree 3J === (used for B/3) (absolute values in followed by sign bits if nonzero)

		0				1				2				3
		0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3
0	0	2	5	8	10	5	6	8	9	7	8	10	10	10	11	11	11
	1	5	6	8	10	6	6	8	10	8	8	8	10	10	9	10	10
	2	8	8	10	10	8	8	10	10	10	9	10	10	10	10	10	10
	3	10	10	11	11	10	9	10	10	11	10	10	10	11	10	10	10
1	0	5	6	8	9	6	6	8	10	8	8	10	10	10	10	10	10
	1	6	6	8	10	6	6	7	9	8	7	9	9	10	9	9	9
	2	8	7	10	10	8	7	9	9	10	8	9	9	10	9	9	9
	3	10	10	10	10	9	9	9	9	10	9	9	9	10	9	9	9
2	0	8	8	9	11	8	7	9	9	10	9	10	10	11	10	10	10
	1	8	7	10	10	7	8	9	9	9	9	9	9	10	9	9	9
	2	9	9	10	10	9	8	9	9	9	9	9	9	10	9	9	9
	3	10	10	10	10	10	9	9	9	10	9	9	9	10	9	9	9
3	0	9	10	10	11	9	9	10	10	10	10	10	10	12	10	10	10
	1	10	10	10	10	9	8	9	9	10	9	9	9	10	9	9	9
	2	12	10	10	10	10	9	9	9	10	9	9	9	10	9	9	9
	3	11	10	10	10	10	9	9	9	11	9	9	9	11	9	9	9

=== Tree 3K === (used for B/4)

	-3	-2	-1	0	1	2	3
-3	9	8	7	7	7	8	9
-2	8	8	7	6	6	8	8
-1	7	6	5	3	5	6	7
0	7	6	4	2	4	6	7
1	7	6	5	4	5	7	7
2	8	8	6	6	7	8	8
3	9	8	7	7	7	8	9

=== Tree 3L === (used for B/5) * -3,-2: 5 bits * -1: 2 bits * 0: 1 bits * 1: 3 bits * 2,3: 5 bits === Tree 3M === (used for B/6) * -3,-2: 5 bits * -1: 3 bits * 0: 1 bits * 1: 2 bits * 2,3: 5 bits === Tree 3N === (used for B/7) (absolute values in followed by sign bits if nonzero)

		0				1				2				3
		0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3
0	0	1	5	9	9	6	7	9	9	10	10	11	11	11	10	10	10
	1	6	8	10	10	7	7	9	9	9	9	10	10	11	10	10	10
	2	10	11	10	11	10	9	10	10	11	10	10	10	11	10	10	10
	3	11	11	11	11	10	10	10	10	11	10	10	10	10	10	10	10
1	0	6	7	10	11	7	7	9	10	10	10	9	10	11	10	10	10
	1	6	7	9	10	7	7	8	9	9	9	9	9	10	9	9	9
	2	10	10	10	10	9	9	9	9	10	9	9	9	10	9	9	9
	3	10	9	10	10	10	9	9	9	10	9	9	9	10	9	9	9
2	0	9	9	11	11	11	9	10	10	10	10	10	10	11	10	10	10
	1	9	9	10	10	9	9	9	9	9	9	9	9	10	9	9	9
	2	10	10	10	10	9	9	9	9	10	8	9	9	10	9	9	9
	3	10	10	10	10	10	9	9	9	10	9	9	9	10	9	9	9
3	0	8	11	11	11	10	10	10	10	11	10	10	10	11	10	10	10
	1	9	9	10	10	10	9	9	9	10	9	9	9	10	9	9	9
	2	10	10	10	10	10	9	9	9	10	9	9	9	10	9	9	9
	3	10	10	10	10	10	9	9	9	10	9	9	9	10	9	9	9

==== Trees for Precision 4 ==== (i.e. symbol set -5..+5) === Tree 4A === (used for A/0) (absolute values in followed by sign bits if nonzero) * 0: 3 bits * 1,2: 2 bits * 3..5: 3 bits === Tree 4B === (used for A/1)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	11	10	9	8	8	7	8	9	9	10	11
-4	9	10	10	9	9	8	9	9	10	10	10
-3	9	10	10	9	8	7	8	9	10	10	10
-2	9	10	9	8	7	6	7	8	9	9	8
-1	8	9	8	7	5	4	5	7	8	9	8
0	7	8	7	6	4	2	4	6	7	8	7
1	8	8	8	7	5	4	6	7	8	9	8
2	9	10	9	8	7	6	7	8	9	9	8
3	9	10	10	9	8	7	8	9	10	10	9
4	10	11	10	9	9	8	9	9	10	10	10
5	11	10	9	9	8	7	8	9	9	10	10

=== Tree 4C === (used for A/2, B/5) (absolute values in followed by sign bits if nonzero) * 0..2: 2 bits * 3: 3 bits * 4,5: 4 bits === Tree 4D === (used for A/3) (absolute values in followed by sign bits if nonzero) * 0: 2 bits * 1: 1 bits * 2: 3 bits * 3: 4 bits * 4,5: 5 bits === Tree 4E === (used for A/4, B/2)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	10	9	9	8	8	8	8	8	8	9	10
-4	10	10	9	9	8	8	8	8	9	9	9
-3	9	9	9	8	7	7	7	7	8	9	8
-2	8	9	8	7	6	6	6	7	8	8	8
-1	8	8	7	6	5	4	5	6	7	8	8
0	8	8	7	6	4	4	4	6	7	8	7
1	8	8	7	6	5	4	5	6	8	8	8
2	8	8	8	7	6	6	6	7	8	9	9
3	8	9	8	8	7	7	7	8	9	10	9
4	9	10	9	8	8	8	8	9	10	10	10
5	10	9	8	8	8	8	8	9	9	10	10

=== Tree 4F === (used for A/5)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	9	9	9	9	8	9	9	9	9	8	8
-4	9	7	9	9	9	7	8	9	9	9	9
-3	8	9	8	9	7	7	7	7	8	9	9
-2	9	9	9	8	6	6	7	6	7	9	7
-1	8	8	8	7	5	4	4	5	6	9	9
0	9	8	7	5	5	4	5	6	7	7	9
1	9	7	7	6	5	4	5	6	9	9	9
2	8	7	8	9	6	6	6	7	9	9	8
3	9	8	8	7	6	6	9	9	7	8	9
4	9	9	9	9	9	9	9	8	9	9	9
5	9	9	9	9	8	9	9	9	9	8	9

=== Tree 4G === (used for A/6)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	8	8	8	8	8	8	8	7	8	8	8
-4	8	8	8	8	7	7	8	8	8	8	8
-3	7	8	7	8	7	8	7	7	8	8	8
-2	8	8	7	6	7	5	6	7	8	7	8
-1	9	8	6	5	6	7	7	7	7	8	9
0	8	7	7	5	5	4	5	5	6	7	8
1	8	8	7	7	6	6	6	6	6	8	8
2	8	8	7	7	6	5	6	7	7	7	8
3	7	7	8	7	7	7	7	7	8	8	8
4	8	8	8	7	8	8	8	8	8	8	8
5	8	8	8	8	7	7	8	8	8	8	8

=== Tree 4H === (used for A/7)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	12	11	11	10	10	9	9	10	10	11	11
-4	11	11	11	10	10	9	10	10	10	11	10
-3	11	11	10	10	9	9	9	9	10	11	10
-2	10	10	10	9	8	7	8	8	9	10	10
-1	10	10	9	8	6	4	6	8	9	10	9
0	9	9	9	7	4	1	4	7	9	10	9
1	9	10	9	8	6	4	6	8	9	10	10
2	10	10	9	8	7	7	8	9	10	10	10
3	10	10	10	9	9	9	9	10	10	11	11
4	11	11	10	10	9	10	10	10	11	12	11
5	11	10	10	9	9	9	10	10	11	11	11

=== Tree 4I === (used for B/0, B/3)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	11	10	10	9	9	8	8	9	9	10	10
-4	10	10	10	9	9	8	9	9	9	10	9
-3	10	10	9	9	8	8	8	8	9	10	9
-2	9	9	9	8	7	6	7	7	8	9	9
-1	9	9	8	7	5	4	5	7	8	9	8
0	8	8	8	6	4	2	4	6	8	9	8
1	8	9	8	7	5	4	5	7	8	9	9
2	9	9	8	7	6	6	7	8	9	9	9
3	9	9	9	8	8	8	8	9	9	10	10
4	10	10	9	9	8	9	9	9	10	11	10
5	10	9	9	8	8	8	9	9	10	10	10

=== Tree 4J === (used for B/1, B/4)

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	12	11	10	9	9	8	9	10	10	11	12
-4	10	11	11	10	10	9	10	10	11	11	11
-3	10	11	11	10	9	8	9	10	11	11	11
-2	10	11	10	9	8	7	8	9	10	10	9
-1	9	10	9	8	6	4	6	8	9	10	9
0	8	9	8	7	4	1	4	7	8	9	8
1	9	9	9	8	6	4	7	8	9	10	9
2	10	11	10	9	8	7	8	9	10	10	9
3	10	11	11	10	9	8	9	10	11	11	10
4	11	12	11	10	10	9	10	10	11	11	11
5	12	11	10	10	9	8	9	10	10	11	11

=== Tree 4K === (used for B/6) (absolute values in followed by sign bits if nonzero) * 0: 2 bits * 1: 1 bits * 2: 3 bits * 3,4: 5 bits * 5: 4 bits === Tree 4L === (used for B/7) * -5,-4: 6 bits * -3: 5 bits * -2: 4 bits * -1: 3 bits * 0: 1 bits * 1: 3 bits * 2: 4 bits * 3,4: 6 bits * 5: 5 bits ==== Trees for Precision 5 ==== (i.e. symbol set -7..+7) === Tree 5A === (used for A/0)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	12	11	11	10	10	10	9	9	9	9	10	10	10	11	11
-6	11	11	11	10	10	9	9	9	9	9	10	10	10	11	11
-5	11	11	11	11	10	10	9	9	9	10	10	10	11	10	10
-4	10	10	11	10	9	9	9	8	8	9	9	10	10	10	10
-3	10	10	10	9	9	8	8	7	8	8	9	9	10	10	10
-2	10	9	10	9	8	7	7	6	6	7	8	9	10	9	9
-1	9	9	9	9	8	7	5	5	5	6	8	9	9	9	9
0	9	9	9	8	7	6	4	3	4	6	7	8	9	9	9
1	9	9	9	8	8	6	5	4	5	7	8	9	9	9	9
2	9	9	10	9	8	7	6	6	7	7	8	9	10	9	10
3	10	10	10	9	9	8	8	7	8	8	9	10	10	10	10
4	10	10	10	10	9	9	8	8	9	9	9	10	11	10	10
5	11	10	11	10	10	10	9	9	9	10	10	11	11	11	11
6	11	11	10	10	10	9	9	9	9	9	10	10	11	11	11
7	11	11	10	10	10	9	9	9	9	10	10	10	11	11	12

=== Tree 5B === (used for A/1, B/1)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	11	10	10	9	9	9	9	8	9	9	9	10	10	10	11
-6	10	10	10	9	9	9	8	8	8	9	9	9	9	10	10
-5	10	10	10	9	9	9	9	9	9	9	9	10	10	10	10
-4	9	9	9	9	9	8	8	8	8	8	9	9	10	9	9
-3	9	9	9	9	8	8	7	7	7	8	8	9	9	9	9
-2	9	9	9	8	8	7	6	6	6	7	8	8	9	9	9
-1	9	9	9	8	7	6	5	5	5	6	7	8	9	9	9
0	8	8	9	8	7	6	5	5	5	6	7	8	9	8	8
1	9	9	9	8	7	6	5	5	5	6	7	8	9	9	9
2	9	9	9	8	8	7	6	6	6	7	8	8	9	9	9
3	9	9	10	9	8	8	7	7	7	8	8	9	9	9	9
4	10	9	10	9	9	8	8	8	8	8	9	9	10	9	10
5	10	10	10	10	9	9	9	9	9	9	9	10	10	10	10
6	10	10	10	9	9	9	8	8	8	9	9	9	10	10	10
7	11	10	10	9	9	9	9	8	9	9	9	10	10	10	11

=== Tree 5C === (used for A/2) (absolute values in followed by sign bits if nonzero) * 0: 3 bits * 1: 2 bits * 2..5: 3 bits * 6,7: 4 bits === Tree 5D === (used for A/3)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	10	10	10	10	10	10	9	9	10	9	10	9	9	9	10
-6	10	10	10	10	9	9	9	9	10	9	9	9	10	10	10
-5	10	9	10	10	10	9	9	9	10	9	10	10	10	10	10
-4	10	10	10	9	9	10	9	9	9	9	9	9	9	10	10
-3	10	10	9	9	10	9	9	8	9	8	8	9	10	9	10
-2	10	10	10	9	9	7	7	6	7	7	7	9	10	9	10
-1	10	10	10	10	8	7	5	4	6	7	8	10	10	10	9
0	9	10	9	10	7	6	4	2	5	6	7	7	9	9	9
1	9	9	10	9	9	7	6	5	6	7	9	9	10	10	10
2	10	9	10	9	8	7	7	7	7	7	10	9	10	10	10
3	10	10	10	10	10	9	8	7	7	9	9	8	10	10	10
4	10	10	9	10	9	10	9	8	10	9	9	10	10	10	10
5	10	10	10	10	10	9	10	9	9	10	10	9	10	10	10
6	10	9	10	9	9	10	9	10	9	10	9	10	9	10	10
7	10	10	10	9	10	10	9	10	10	10	10	10	9	10	10

=== Tree 5E === (used for A/4, B/4) (absolute values in followed by sign bits if nonzero) * 0: 3 bits * 1,2: 2 bits * 3: 3 bits * 4..7: 4 bits === Tree 5F === (used for A/5, B/5) (absolute values in followed by sign bits if nonzero) * 0: 1 bits * 1: 2 bits * 2: 3 bits * 3: 5 bits * 4,5: 6 bits * 6,7: 5 bits === Tree 5G === (used for A/6) * -7: 7 bits * -6: 6 bits * -5: 7 bits * -4: 6 bits * -3: 5 bits * -2: 4 bits * -1..1: 2 bits * 2,3: 5 bits * 4: 6 bits * 5: 7 bits * 6: 6 bits * 7: 7 bits === Tree 5H === (used for A/7) * -7: 6 bits * -6: 5 bits * -5: 6 bits * -4: 5 bits * -3: 4 bits * -2,-1: 3 bits * 0: 2 bits * 1: 3 bits * 2,3: 4 bits * 4: 5 bits * 5: 6 bits * 6: 5 bits * 7: 6 bits === Tree 5I === (used for B/0, B/2)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	11	11	10	10	10	9	9	8	9	9	9	10	10	10	11
-6	11	10	10	10	10	9	9	8	9	9	10	10	10	10	11
-5	10	10	11	11	10	10	9	9	9	10	10	11	11	10	10
-4	10	10	11	10	10	9	9	8	9	9	10	10	11	10	10
-3	9	10	10	10	9	9	8	7	8	8	9	10	10	10	9
-2	9	9	10	9	9	8	7	6	7	8	8	9	10	9	9
-1	9	9	9	9	8	7	5	4	5	7	8	9	9	9	9
0	8	8	9	8	7	6	4	3	4	6	7	8	9	8	8
1	9	9	9	9	8	7	5	4	5	7	8	9	9	9	9
2	9	9	10	9	8	8	7	6	7	8	8	9	10	9	9
3	9	9	10	10	9	8	8	7	8	8	9	10	10	10	10
4	10	10	10	10	10	9	9	8	9	9	10	10	11	10	10
5	10	10	11	11	10	10	9	9	9	10	10	11	11	10	10
6	11	10	10	10	9	9	9	8	9	9	10	10	10	10	11
7	11	11	10	10	9	9	9	8	9	9	10	10	10	11	11

=== Tree 5J === (used for B/3)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	12	11	11	10	10	10	10	9	9	10	10	12	11	12	12
-6	12	12	12	11	11	10	9	9	9	10	11	11	12	11	12
-5	11	11	12	12	11	11	10	10	10	11	11	12	12	11	12
-4	11	11	12	11	11	10	10	9	9	10	11	12	12	11	10
-3	11	11	11	11	10	9	9	8	8	9	10	10	11	10	10
-2	10	10	11	10	9	8	7	6	8	8	9	10	10	10	9
-1	9	9	11	9	8	7	6	5	6	8	9	10	10	9	9
0	9	9	9	9	8	7	5	1	5	7	8	9	10	9	9
1	9	10	10	9	9	8	6	5	6	7	9	10	10	10	9
2	10	10	11	10	9	8	7	6	8	8	9	11	11	10	10
3	10	10	11	11	10	9	9	8	9	10	10	11	12	10	10
4	11	11	12	12	10	10	10	9	10	10	10	11	12	11	11
5	11	11	12	12	12	11	10	9	10	11	11	12	12	11	11
6	11	11	11	11	10	10	10	9	10	10	11	11	12	11	11
7	12	11	11	11	11	10	9	9	9	10	10	10	11	12	12

=== Tree 5K === (used for B/6)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	12	11	10	11	10	10	10	9	9	10	9	11	11	10	12
-6	10	11	11	10	10	10	9	8	10	10	10	11	11	11	11
-5	11	11	12	12	11	10	10	10	10	11	10	11	12	10	11
-4	11	11	11	11	12	10	10	9	9	10	11	11	12	10	11
-3	10	10	11	11	9	9	9	8	10	10	11	10	11	10	11
-2	9	10	11	10	11	10	8	7	8	9	10	10	11	10	9
-1	9	12	11	11	10	8	7	5	7	8	9	10	10	10	12
0	9	9	10	9	8	7	4	1	5	7	9	10	10	10	9
1	9	10	10	9	9	8	6	4	6	8	10	10	11	10	9
2	9	10	10	10	10	9	8	7	8	10	9	10	11	11	10
3	11	10	11	10	10	10	10	9	9	10	10	11	12	9	10
4	11	11	12	12	10	10	11	9	10	10	10	10	11	12	10
5	10	10	11	12	12	11	11	10	11	10	12	10	12	10	11
6	11	10	10	10	10	9	10	9	10	10	11	10	11	11	10
7	12	10	12	12	10	9	9	8	10	10	10	10	11	10	11

=== Tree 5L === (used for B/7)

	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7
-7	12	12	11	11	11	11	10	9	10	10	11	12	11	12	12
-6	12	12	12	12	11	11	10	10	10	10	11	12	12	11	12
-5	11	11	12	12	12	11	11	10	11	11	12	12	12	12	12
-4	11	11	12	12	12	11	10	9	10	10	12	12	12	11	11
-3	11	11	11	12	11	10	9	8	9	10	10	11	12	11	11
-2	10	11	12	11	10	9	8	7	8	9	10	10	11	11	10
-1	10	10	12	10	9	8	7	3	7	8	9	10	11	10	10
0	9	9	10	9	8	7	3	2	3	7	9	9	10	9	9
1	10	10	11	10	9	8	7	3	7	8	9	10	11	10	10
2	10	10	11	11	10	9	8	7	8	9	10	11	11	10	10
3	11	11	11	11	10	10	9	8	9	10	11	12	12	11	11
4	12	11	12	12	11	11	10	9	10	11	11	12	12	12	11
5	11	12	12	12	12	11	11	10	11	11	12	12	12	11	12
6	12	12	12	12	11	10	10	9	10	10	11	12	12	12	12
7	12	12	12	11	11	10	10	9	10	10	11	11	12	12	12

==== Trees for Precision 6 ==== (i.e. symbol set -15..+15) === Tree 6A === (used for A/0, B/0) (absolute values in followed by sign bits if nonzero) * 0: 3 bits * 1: 2 bits * 2,3: 3 bits * 4,5: 4 bits * 6..9: 5 bits * 10..12: 6 bits * 13,14: 5 bits * 15: 6 bits === Tree 6B === (used for A/1, B/4) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	2	4	6	7	8	9	9	10	10	10	11	12	10	9	9	9
1	3	4	5	6	7	8	8	9	9	10	10	10	10	8	8	9
2	6	5	6	7	8	8	9	9	10	10	10	11	10	8	9	9
3	7	6	7	8	8	9	9	10	10	10	10	11	10	8	9	9
4	8	7	7	8	9	9	10	10	10	10	10	11	10	9	9	9
5	8	8	8	9	9	9	10	10	10	11	11	11	11	9	9	9
6	9	8	9	9	10	10	10	11	11	11	11	11	10	9	10	10
7	9	9	9	9	10	10	11	11	11	11	12	12	11	9	10	10
8	10	9	9	10	10	10	10	11	11	11	12	12	11	10	10	10
9	10	10	10	10	11	10	11	11	11	11	12	12	11	10	10	10
10	11	10	10	10	11	11	11	11	12	12	12	12	11	10	10	10
11	11	10	10	11	11	11	11	12	12	12	12	12	11	10	10	10
12	10	9	10	10	11	11	11	11	11	11	11	11	11	10	10	11
13	9	8	8	9	9	9	9	10	9	10	10	10	10	11	11	11
14	9	8	8	9	9	9	10	10	10	10	10	10	10	11	11	11
15	9	9	9	9	10	9	10	10	10	11	10	11	11	11	11	11

=== Tree 6C === (used for A/2) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	4	5	6	6	7	8	8	9	9	9	10	10	10	9	9	10
1	4	4	5	6	6	7	7	8	8	9	9	9	9	8	9	9
2	6	5	5	6	7	7	8	8	8	9	9	10	9	8	9	9
3	6	6	6	6	7	7	8	8	9	9	9	10	9	8	9	9
4	7	6	6	7	7	8	8	9	9	9	9	10	9	9	9	9
5	8	7	7	7	8	8	8	9	9	9	10	10	10	9	9	10
6	8	7	7	8	8	8	9	9	9	10	10	10	10	9	9	10
7	9	8	8	8	8	9	9	9	10	10	10	10	10	9	10	10
8	9	8	8	8	9	9	9	10	10	10	10	11	10	9	10	10
9	9	8	9	9	9	9	9	10	10	10	11	11	10	10	10	10
10	10	9	9	9	9	10	10	10	10	11	11	11	11	10	10	10
11	10	9	9	10	10	10	10	10	10	11	11	11	11	10	11	11
12	10	9	9	9	9	10	10	10	10	10	11	11	11	10	11	11
13	9	8	8	8	9	9	9	9	9	10	10	10	10	11	11	11
14	9	8	9	9	9	9	9	10	10	10	10	10	11	11	11	11
15	10	9	9	9	9	9	10	10	10	10	10	11	11	11	11	11

=== Tree 6D === (used for A/3) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	3	4	5	7	7	8	9	9	10	10	10	11	10	9	10	10
1	4	4	5	6	6	7	8	8	9	9	9	10	9	8	9	9
2	5	5	5	6	7	8	8	8	9	9	10	10	9	8	9	9
3	7	6	6	7	7	8	8	9	9	10	10	10	10	8	9	9
4	7	7	7	7	8	8	8	9	9	10	10	10	9	9	9	9
5	8	7	7	8	8	8	9	9	9	10	10	10	10	9	9	10
6	9	8	8	8	8	9	9	10	10	10	10	11	10	9	9	10
7	9	8	8	9	9	9	9	10	10	10	10	11	10	9	9	10
8	10	9	9	9	9	9	10	10	10	10	11	11	10	9	10	10
9	10	9	9	10	10	10	10	10	10	11	11	11	11	10	10	10
10	10	10	9	10	10	10	10	11	11	11	11	11	11	10	10	10
11	11	10	10	10	10	10	11	11	11	11	12	12	11	10	10	10
12	10	9	9	10	10	10	10	10	10	10	11	11	11	10	10	11
13	9	8	8	8	9	9	9	9	9	9	10	10	10	10	10	11
14	10	9	9	9	9	9	9	9	10	10	10	10	11	11	11	11
15	10	9	9	9	10	10	10	10	10	10	10	11	11	11	11	11

=== Tree 6E === (used for A/4) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	6	7	8	6	7	9	9	9	9	9	9	7	9	9	9	9
1	7	7	6	6	6	8	6	9	7	8	9	9	9	8	9	9
2	5	6	6	7	6	7	6	7	9	7	9	9	9	9	9	8
3	7	7	8	7	6	6	7	9	8	8	9	9	9	9	9	9
4	8	7	6	7	8	7	7	7	8	8	9	9	9	9	9	9
5	7	7	7	7	6	7	7	8	8	9	9	9	8	9	9	9
6	7	9	6	8	7	7	8	9	9	8	9	8	9	9	9	9
7	8	9	6	8	9	7	7	9	9	8	9	9	9	9	9	9
8	8	9	8	8	8	9	8	9	7	9	9	9	9	9	9	9
9	8	9	9	9	9	9	8	8	9	9	9	9	9	9	9	9
10	8	9	8	9	9	9	9	9	9	9	9	9	9	9	9	9
11	9	8	9	9	9	9	9	9	9	8	9	9	9	9	9	9
12	9	9	8	9	9	9	9	9	9	9	9	9	9	9	9	9
13	9	9	9	9	8	9	9	9	9	9	9	9	9	9	9	9
14	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9
15	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9

=== Tree 6F === (used for A/5) * -15,-14: 7 bits * -13: 6 bits * -12: 7 bits * -11..-9: 6 bits * -8,-7: 5 bits * -6: 6 bits * -5: 5 bits * -4..3: 4 bits * 4: 3 bits * 5..7: 5 bits * 8,9: 6 bits * 10..13: 7 bits * 14: 6 bits * 15: 7 bits === Tree 6G === (used for A/6, B/7) * -15: 5 bits * -14: 10 bits * -13,-12: 8 bits * -11..-9: 7 bits * -8..-5: 6 bits * -4,-3: 5 bits * -2: 4 bits * -1: 3 bits * 0: 2 bits * 1: 3 bits * 2: 4 bits * 3,4: 5 bits * 5..8: 6 bits * 9..11: 7 bits * 12: 8 bits * 13: 9 bits * 14: 10 bits * 15: 5 bits === Tree 6H === (used for A/7) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	3	4	5	6	7	8	8	9	9	10	10	10	10	8	9	9
1	4	4	5	6	6	7	7	8	8	9	9	10	9	8	8	8
2	5	5	6	7	7	8	8	8	9	9	10	10	9	8	9	9
3	6	6	6	7	8	8	8	9	9	9	10	10	10	8	9	9
4	7	6	7	8	8	8	9	9	9	10	10	10	10	9	9	9
5	8	7	8	8	8	9	9	10	10	10	10	11	10	9	9	9
6	8	7	8	8	9	9	9	10	10	10	11	11	11	9	10	10
7	9	8	8	9	9	9	10	10	10	11	11	12	11	10	10	10
8	9	8	9	9	9	10	10	10	10	11	11	11	11	10	10	10
9	10	9	9	9	10	10	10	11	11	11	11	11	11	10	10	10
10	10	9	9	10	10	10	11	11	11	11	11	12	11	10	10	11
11	10	9	10	10	10	11	11	11	11	12	12	12	11	10	11	11
12	10	9	9	9	10	10	10	11	11	11	11	11	12	10	11	11
13	8	8	8	8	9	9	9	10	10	10	10	10	10	11	11	11
14	9	8	8	9	9	9	10	10	10	10	10	10	11	11	11	11
15	9	8	9	9	9	10	10	10	10	10	10	11	11	11	12	12

=== Tree 6I === (used for B/1) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	4	4	5	6	7	8	8	8	9	9	10	10	10	9	9	9
1	4	4	5	6	6	7	7	8	8	9	9	9	9	8	8	9
2	5	5	6	6	7	7	8	8	8	9	9	9	9	8	9	9
3	6	6	6	7	7	8	8	8	9	9	9	10	9	8	9	9
4	7	6	7	7	8	8	9	9	9	9	10	10	10	9	9	9
5	8	7	7	8	8	8	9	9	9	10	10	10	10	9	9	10
6	8	7	8	8	9	9	9	9	10	10	10	10	10	9	9	10
7	8	8	8	8	9	9	9	10	10	10	10	11	10	9	10	10
8	9	8	8	9	9	9	10	10	10	10	11	11	11	10	10	10
9	9	8	9	9	9	10	10	10	10	11	11	11	11	10	10	10
10	9	9	9	9	10	10	10	10	11	11	11	11	11	10	10	11
11	10	9	9	10	10	10	11	11	11	11	11	11	11	10	10	11
12	9	9	9	9	10	10	10	10	10	11	11	11	11	10	11	11
13	9	8	8	8	9	9	9	9	10	10	10	10	10	11	11	11
14	9	8	8	9	9	9	9	10	10	10	10	10	11	11	11	11
15	9	8	9	9	9	10	10	10	10	10	11	11	11	11	12	12

=== Tree 6J === (used for B/2, B/3) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	3	4	5	6	7	8	9	9	9	10	10	11	10	9	9	9
1	3	4	5	6	7	7	8	8	9	9	9	10	9	8	8	9
2	5	5	6	7	7	8	8	9	9	9	10	10	9	8	9	9
3	6	6	7	7	8	8	9	9	10	10	10	10	10	8	9	9
4	7	7	7	8	8	9	9	9	10	10	10	11	10	9	9	9
5	8	7	8	8	9	9	9	10	10	10	11	11	10	9	9	9
6	8	8	8	9	9	9	10	10	10	10	11	11	10	9	9	10
7	9	8	9	9	9	10	10	10	11	11	11	11	11	9	10	10
8	9	8	9	9	10	10	10	10	11	11	11	12	11	10	10	10
9	10	9	9	10	10	10	11	11	11	11	11	12	11	10	10	10
10	10	9	10	10	10	11	11	11	12	11	12	12	11	10	10	11
11	10	9	10	10	11	11	11	11	12	12	12	12	11	10	10	11
12	10	9	9	10	10	10	10	11	11	11	11	11	11	10	11	11
13	9	8	8	8	9	9	9	9	10	10	10	10	10	11	11	11
14	9	8	8	9	9	9	9	10	10	10	10	10	10	11	11	11
15	9	9	9	9	9	10	10	10	10	10	10	11	11	11	11	12

=== Tree 6K === (used for B/5) * -15..-13: 7 bits * -12..-7: 8 bits * -6..-4: 7 bits * -3: 6 bits * -2: 5 bits * -1: 3 bits * 0: 1 bits * 1: 3 bits * 2: 5 bits * 3,4: 6 bits * 5..7: 7 bits * 8,9: 8 bits * 10,11: 9 bits * 12: 8 bits * 13: 6 bits * 14,15: 7 bits === Tree 6L === (used for B/6) (absolute values in followed by sign bits if nonzero)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	1	4	7	8	9	9	9	11	9	12	12	12	10	9	9	10
1	4	4	6	8	8	9	9	10	10	10	11	12	10	8	8	9
2	6	6	7	8	9	8	9	10	10	10	11	10	10	9	9	9
3	8	7	8	9	9	10	10	10	11	10	10	11	9	9	9	10
4	9	9	10	11	10	12	10	11	10	12	10	10	11	9	11	9
5	10	10	9	9	12	11	11	12	11	10	11	12	11	10	11	11
6	10	9	10	10	10	11	11	11	11	10	11	12	11	10	10	10
7	10	9	10	10	12	11	11	11	12	11	11	12	12	10	10	9
8	10	10	10	10	11	12	12	12	11	11	12	12	12	9	10	10
9	10	10	10	12	11	11	12	11	12	11	12	12	12	12	10	10
10	10	10	12	11	11	12	12	12	11	12	12	11	10	10	10	11
11	10	10	11	12	12	12	12	11	12	12	12	12	12	10	10	10
12	11	9	10	9	10	10	12	10	10	12	11	11	11	12	11	12
13	9	9	8	9	10	9	10	11	10	10	11	11	12	11	12	10
14	9	9	9	10	11	10	10	9	10	10	12	11	11	12	10	11
15	10	10	9	10	10	10	10	10	11	11	11	11	10	11	11	11

==== Trees for Precision 7 ==== (i.e. symbol set -31..+31) === Tree 7A === (used for A/0, A/7, B/1, B/2) * -31..-29: 8 bits * -28..-26: 7 bits * -25: 8 bits * -24: 10 bits * -23..-18: 9 bits * -17..-11: 8 bits * -10..-7: 7 bits * -6..-4: 6 bits * -3: 5 bits * -2: 4 bits * -1: 3 bits * 0: 2 bits * 1: 3 bits * 2: 4 bits * 3: 5 bits * 4..6: 6 bits * 7..10: 7 bits * 11..16: 8 bits * 17..23: 9 bits * 24: 10 bits * 25: 8 bits * 26..28: 7 bits * 29..31: 8 bits === Tree 7B === (used for A/1) * -31..-29: 8 bits * -28..-26: 7 bits * -25: 8 bits * -24..-22: 9 bits * -21..-15: 8 bits * -14..-9: 7 bits * -8..-6: 6 bits * -5..-3: 5 bits * -2: 4 bits * -1..1: 3 bits * 2: 4 bits * 3,4: 5 bits * 5..8: 6 bits * 9..14: 7 bits * 15..21: 8 bits * 22..24: 9 bits * 25: 8 bits * 26,27: 7 bits * 28..31: 8 bits === Tree 7C === (used for A/2) * -31..-26: 8 bits * -25: 7 bits * -24..-22: 8 bits * -21..-16: 7 bits * -15: 6 bits * -14,-13: 7 bits * -12,-11: 6 bits * -10..-8: 5 bits * -7: 6 bits * -6..-1: 5 bits * 0: 4 bits * 1: 5 bits * 2,3: 4 bits * 4: 6 bits * 5..7: 5 bits * 8: 6 bits * 9: 5 bits * 10..12: 6 bits * 13: 7 bits * 14: 6 bits * 15..19: 7 bits * 20,21: 8 bits * 22,23: 7 bits * 24: 8 bits * 25,26: 7 bits * 27..29: 8 bits * 30: 7 bits * 31: 8 bits === Tree 7D === (used for A/3) * -31,-30: 8 bits * -29..-26: 7 bits * -25..-17: 8 bits * -16..-11: 7 bits * -10..-7: 6 bits * -6..-4: 5 bits * -3..3: 4 bits * 4..6: 5 bits * 7..10: 6 bits * 11..16: 7 bits * 17..24: 8 bits * 25..30: 7 bits * 31: 8 bits === Tree 7E === (used for A/4) * -31,-30: 9 bits * -29: 8 bits * -28,-27: 7 bits * -26: 8 bits * -25: 7 bits * -24,-23: 8 bits * -22: 7 bits * -21: 8 bits * -20: 9 bits * -19..-13: 8 bits * -12: 7 bits * -11: 6 bits * -10: 5 bits * -9: 6 bits * -8..-3: 5 bits * -2..2: 4 bits * 3..6: 5 bits * 7: 6 bits * 8: 5 bits * 9..14: 6 bits * 15..21: 7 bits * 22: 8 bits * 23,24: 9 bits * 25: 8 bits * 26,27: 9 bits * 28: 8 bits * 29..31: 9 bits === Tree 7F === (used for A/5) * -31..-29: 8 bits * -28..-26: 7 bits * -25: 8 bits * -24..-22: 9 bits * -21..-16: 8 bits * -15..-11: 7 bits * -10..-7: 6 bits * -6..-4: 5 bits * -3..-1: 4 bits * 0: 3 bits * 1,2: 4 bits * 3..6: 5 bits * 7..10: 6 bits * 11..15: 7 bits * 16..21: 8 bits * 22: 9 bits * 23..25: 8 bits * 26..28: 7 bits * 29..31: 8 bits === Tree 7G === (used for A/6) * -31..-27: 8 bits * -26: 7 bits * -25: 8 bits * -24..-20: 9 bits * -19..-13: 8 bits * -12..-8: 7 bits * -7..-5: 6 bits * -4,-3: 5 bits * -2,-1: 4 bits * 0: 2 bits * 1: 3 bits * 2: 4 bits * 3,4: 5 bits * 5..7: 6 bits * 8..13: 7 bits * 14..19: 8 bits * 20..24: 9 bits * 25: 8 bits * 26: 7 bits * 27..31: 8 bits === Tree 7H === (used for B/0, B/5) * -31..-25: 7 bits * -24..-20: 9 bits * -19..-14: 8 bits * -13..-9: 7 bits * -8..-5: 6 bits * -4,-3: 5 bits * -2: 4 bits * -1..1: 3 bits * 2: 4 bits * 3,4: 5 bits * 5..8: 6 bits * 9..12: 7 bits * 13..19: 8 bits * 20..24: 9 bits * 25..31: 7 bits === Tree 7I === (used for B/3) * -31: 9 bits * -30,-29: 8 bits * -28: 9 bits * -27..-25: 8 bits * -24: 11 bits * -23: 10 bits * -22: 11 bits * -21..-16: 10 bits * -15..-10: 9 bits * -9..-7: 8 bits * -6..-4: 7 bits * -3: 6 bits * -2: 5 bits * -1: 3 bits * 0: 1 bits * 1: 3 bits * 2: 5 bits * 3: 6 bits * 4..6: 7 bits * 7..10: 8 bits * 11: 9 bits * 12: 10 bits * 13..15: 9 bits * 16..19: 10 bits * 20: 11 bits * 21: 10 bits * 22..24: 11 bits * 25..29: 8 bits * 30,31: 9 bits === Tree 7J === (used for B/4) * -31..-28: 8 bits * -27,-26: 7 bits * -25: 8 bits * -24..-19: 9 bits * -18..-13: 8 bits * -12..-8: 7 bits * -7..-5: 6 bits * -4,-3: 5 bits * -2,-1: 4 bits * 0: 2 bits * 1: 3 bits * 2: 4 bits * 3,4: 5 bits * 5..7: 6 bits * 8..12: 7 bits * 13..18: 8 bits * 19..24: 9 bits * 25: 8 bits * 26,27: 7 bits * 28..31: 8 bits === Tree 7K === (used for B/6) * -31..-25: 8 bits * -24..-19: 9 bits * -18..-13: 8 bits * -12..-8: 7 bits * -7..-5: 6 bits * -4,-3: 5 bits * -2: 4 bits * -1: 3 bits * 0: 2 bits * 1: 3 bits * 2,3: 5 bits * 4..7: 6 bits * 8..12: 7 bits * 13..18: 8 bits * 19..24: 9 bits * 25..31: 8 bits === Tree 7L === (used for B/7) * -31,-30: 11 bits * -29: 10 bits * -28: 11 bits * -27..-20: 9 bits * -19,-18: 8 bits * -17: 7 bits * -16: 8 bits * -15..-11: 7 bits * -10..-6: 6 bits * -5..-3: 5 bits * -2,-1: 4 bits * 0: 2 bits * 1,2: 4 bits * 3..5: 5 bits * 6..10: 6 bits * 11..16: 7 bits * 17..20: 8 bits * 21..27: 9 bits * 28..30: 10 bits * 31: 11 bits ===== Tree Sets ===== This table shows the mapping of tree set IDs, tree IDs and quantization precisions to the code tables defined above. If the table says //n x Tree XX// it means that n symbols of that tree (which might groups of 1, 2 or 4 quantized coefficients) are clustered together. Each cluster is preceeded by an "data present" bit. If that bit is zero, all coefficients of the whole cluster are zero, and no codes for the cluster are stored. If that bit is one that at least one non-zero entry is present and the codes for this cluster follow as in non-clustered mode. The tree choice can be limited to a restricted tree set. The IDs inside the restricted tree set are in the range 0..3 instead of 0..7, and printed as R0..R3 in the table. ==== Tree Set A==== |Quant\ID^ 0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ ^ R0^ R1^ R2^ R3^ ^ 1| Tree 1A| 4 x Tree 1B| Tree 1C| Tree 1D| Tree 1E| Tree 1F| 2 x Tree 1G| Tree 1H| | Tree 1A| Tree 1F| Tree 1E| 4x Tree 1B| ^ 2| Tree 2A| Tree 2B| Tree 2C| Tree 2D| Tree 2E| Tree 2F| Tree 2G| 2 x Tree 2H| | Tree 2A| Tree 2B| Tree 2C| Tree 2D| ^ 3| Tree 3A| Tree 3B| Tree 3C| Tree 3D| Tree 3E| Tree 3F| Tree 3G| Tree 3H| | Tree 3D| Tree 3A| Tree 3E| Tree 3C| ^ 4| Tree 4A| Tree 4B| Tree 4C| Tree 4D| Tree 4E| Tree 4F| Tree 4G| Tree 4H| | Tree 4E| Tree 4A| Tree 4B| Tree 4C| ^ 5| Tree 5A| Tree 5B| Tree 5C| Tree 5D| Tree 5E| Tree 5F| Tree 5G| Tree 5H| | Tree 5B| Tree 5A| Tree 5E| Tree 5D| ^ 6| Tree 6A| Tree 6B| Tree 6C| Tree 6D| Tree 6E| Tree 6F| Tree 6G| 2 x Tree 6H| | Tree 6D| Tree 6A| Tree 6C| Tree 6B| ^ 7| Tree 7A| Tree 7B| Tree 7C| Tree 7D| Tree 7E| Tree 7F| Tree 7G| 4 x Tree 7A| | Tree 7A| Tree 7D| Tree 7B| Tree 7C| ==== Tree Set B==== |Quant\ID^ 0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ ^ R0^ R1^ R2^ R3^ ^ 1| 4 x Tree 1I| Tree 1C| 4 x Tree 1E| Tree 1F| Tree 1J| 2 x Tree 1G| Tree 1E| Tree 1K| | Tree 1J| 4 x Tree 1I| Tree 1C| 4 x Tree 1E| ^ 2| Tree 2I| Tree 2J| 4 x Tree 2D| Tree 2K| 4 x Tree 2L| Tree 2M| Tree 2N| 4 x Tree 2K| | Tree 2I| Tree 2K| 4 x Tree 2D| Tree 2J| ^ 3| 4 x Tree 3I| Tree 3B| Tree 3A| Tree 3J| Tree 3K| 4 x Tree 3L| Tree 3M| 4 x Tree 3N| | 4 x Tree 3I| Tree 3B| Tree 3A| Tree 3J| ^ 4| Tree 4I| Tree 4J| Tree 4E| 2 x Tree 4I| 4 x Tree 4J| Tree 4C| Tree 4K| Tree 4L| | Tree 4I| Tree 4J| Tree 4E| 4 x Tree 4J| ^ 5| Tree 5I| Tree 5B| 2 x Tree 5I| Tree 5J| Tree 5E| Tree 5F| 4 x Tree 5K| Tree 5L| | Tree 5I| Tree 5B| 2 x Tree 5I| Tree 5J| ^ 6| Tree 6A| Tree 6I| 2 x Tree 6J| Tree 6J| Tree 6B| Tree 6K| Tree 6L| Tree 6G| | Tree 6I| Tree 6B| 2 x Tree 6J| Tree 6A| ^ 7| Tree 7H| Tree 7A| 4 x Tree 7A| Tree 7I| Tree 7J| 2 x Tree 7H| Tree 7K| Tree 7L| | Tree 7H| Tree 7A| 4 x Tree 7A| Tree 7I|