Channel Codes

About File Formats for Parity-Check Matrices

Currently, two different ASCII file formats for storing parity-check matrices are widely used:

  • The so-called AList format was invited by David MacKay et al. and is especially suitable for LDPC codes.
  • The straightforward approach is to store ASCII characters "0" and "1", using whitespaces to separate entries within one row and the newline character to separate rows.

Our decoding toolbox contains flexible parsers and formatters (written in Python) for both matrix storage formats in a submodule that can easily be extracted to be used stand-alone.

The AList Format

The AList format is based on explicit enumeration of non-zero indices in the matrix. It redundantly contains both column-based and row-based indices. All matrix indices are 1-based.


Specifically, the first four lines are as follows:

n m

dvmax dcmax

dv1 dv2 ... dvn

dc1 dc2 ... dc

with

  • n  number of variable nodes (columns)
  • m number of check nodes (rows)
  • dvmax maximum variable node degree (i.e. max column weight)
  • dcmax maximum check node degree (i.e. max row weight)
  • dvi  i-th variable node degree (column weight)
  • dcj j-th check node degree (row weight)

The next n lines contain a list of non-zero entries of the columns, which is right-padded with zeros such that the number of entries is dvmax for each of these lines. Afterwards, m lines are created in the same way for the non-zero row indices.


For example, the 3×7 parity-check matrix of the binary (7,4) Hamming code is represented by

 

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

For non-binary alists the format is very similar to the format for binary codes. However the Galois field size and the value of the non-binary matrix entries need to be specified:

  • In the first row a third value is introduced for the size of the Galois field q, e.g. 8 4 64 (8 symbols, 4 parity symbols, GF size 64).
  • After each position entry its additional non-binary entry value is specified. The values are specified as powers of the primitive element, see here

Example: 16 code symbols, 8 parity symbols, GF size q=64. First column has entries in row 4 and 6, with non-binary values (powers) of 32 and 39.

 

16 8 64
2 4
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
4 4 4 4 4 4 4 4
4 32 6 39
3 6 7 36
5 52 8 62
1 27 2 60
2 45 7 45
3 21 6 2
1 1 3 56 
... 

The 0/1 Format

The 0/1 format simply contains ASCII 0 and 1 entries separated by whitespaces and newlines. Parsers should try to be flexible about the number and type of whitespaces (tabs, spaces) and the newline characters (Windows, UNIX).

The above Hamming code matrix in 0/1 format is

 

1 0 0 1 1 0 1 
0 1 0 1 0 1 1 
0 0 1 0 1 1 1 
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