Elias delta coding

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Elias delta code is a universal code encoding the positive integers. To code a number:

  1. Write it in binary.
  2. Count the bits, remove the leading one, and write that number of starting bits in binary preceding the previous bit string.
  3. Subtract 1 from the number of bits written in step 2 and prepend that many zeros.

An equivalent way to express the same process:

  1. Separate the integer into the highest power of 2 it contains (2N' ) and the remaining N' binary digits of the integer.
  2. Encode N = N' + 1 with Elias gamma coding.
  3. Append the remaining N' binary digits to this representation of N.

The code begins:

 1 = 20 => N' = 0, N = 1 => 1
 2 = 21 + 0 => N' = 1, N = 2 => 0100
 3 = 21 + 1 => N' = 1, N = 2 => 0101
 4 = 22 + 0 => N' = 2, N = 3 => 01100
 5 = 22 + 1 => N' = 2, N = 3 => 01101
 6 = 22 + 2 => N' = 2, N = 3 => 01110
 7 = 22 + 3 => N' = 2, N = 3 => 01111
 8 = 23 + 0 => N' = 3, N = 4 => 00100000
 9 = 23 + 1 => N' = 3, N = 4 => 00100001
10 = 23 + 2 => N' = 3, N = 4 => 00100010
11 = 23 + 3 => N' = 3, N = 4 => 00100011
12 = 23 + 4 => N' = 3, N = 4 => 00100100
13 = 23 + 5 => N' = 3, N = 4 => 00100101
14 = 23 + 6 => N' = 3, N = 4 => 00100110
15 = 23 + 7 => N' = 3, N = 4 => 00100111
16 = 24 + 0 => N' = 4, N = 5 => 001010000
17 = 24 + 1 => N' = 4, N = 5 => 001010001

To decode an Elias delta-coded integer:

  1. Read and count zeroes from the stream until you reach the first one. Call this count of zeroes L.
  2. Considering the one that was reached to be the first digit of an integer, with a value of 2L, read the remaining L digits of the integer. Call this integer M.
  3. Put a one in the first place of our final output, representing the value 2M. Read and append the following M digits.

Example:

001010001
1. 2 leading zeros in 001
2. read 2 more bits i.e. 00101
3. decode N = 00101 = 5
4. get N' = 5 - 1 = 4 remaining bits for the complete code i.e. '0001'
5. encoded number = 24 + 1 = 17

This code can be generalized to zero or negative integers in the same ways described in Elias gamma coding.

See also: Elias gamma coding, Elias omega coding

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