TBE

Truncated Binary Encoding

https://en.wikipedia.org/wiki/Truncated_binary_encoding

k = floor_log2(n);
u = 2 * exp2(k) - n

Writing

if x < u write k least significant bits of x else add u to x and write k most significant bits of x and then write a least significant bit of x.

Reading

1. read k bits as x.
2. if u <= x then read an additional bit and add it to x as a least significant bit and substract u.

Examples (MSB first)

n = 2

k = 1;
u = 4 - 2 = 2;

BE	TBE
0	0
1	1
n		u

n = 3

k = 2;
u = 1

BE	TBE(MSB)	TBE(LSB)
00	0X	0X
01	10	10	u
10	11	11

n = 4

k = 2
u = 8 - 4 = 4

BE	TBE
00	00
01	01
10	10
11	11
n		u

n = 5

k = 2;
u = 3;

BE	TBE
000	00
001	01
010	10
011	110	u
100	111

n = 6

k = 2;
u = 2;

BE	TBE
000	00
001	01
010	100
011	101
100	110
101	111

n = 7

BE	TBE
000	00
001	010
010	011
011	100
100	101
101	110
110	111

n = 10

k = 3;
u = 6;

BE	TBE(MSB)	TBE(LSB)
0000	000X	000X
0001	001X	100X
0010	010X	010X
0011	011X	110X
0100	100X	001X
0101	101X	101X
0110	1100	0110
0111	1101	0111
1000	1110	1110
1001	1111	1111