Linear Dependence in Parity Check Matrices

If a code is specified by means of a M by N parity check matrix, H, in which some rows are linearly dependent - a situation that is usually avoided - it would be possible to map more than the usual K=N-M message bits into a codeword, since one or more rows of H could have been deleted without affecting which bit vectors are codewords.

However, this software does not increase the number of message bits in this case, but instead produces a generator matrix in which some rows are all zero, which will cause some bits of the codeword to always be zero, regardless of the source message. Referring to the description of generator matrix representations, this is accomplished by continuing to compute what would normally become A^-1 (for a dense or mixed representations) or the L and U matrices (for a sparse representation), even though singularity has become apparent.

Example: The parity check matrix created below is redundant, since the 10100 row is equal to the sum of the 11000 and 01100 rows.

make-pchk dep.pchk 4 5 0:0 0:1 1:1 1:2 2:0 2:2 3:3 3:4
print-pchk -d dep.pchk

Parity check matrix in dep.pchk (dense format):

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

make-gen dep.pchk dep.gen dense
Note: Parity check matrix has 1 redundant checks
Number of 1s per check in Inv(A) X B is 0.2
print-gen dep.gen

Generator matrix (dense representation):

Column order (message bits at end):

   0   1   2   3   4

Inv(A) X B:

 0
 0
 0
 1

The generator matrix above can be used to encode message blocks containing one bit. This message bit is copied unchanged to the last bit (numbered 4) of the codeword, and the first four bits of the codeword are set by multiplying A^-1B, shown above, by the message bit. The result is that the first three bits of the codeword produced are always zeros, and the last two bits are always the same (and equal to the message bit).

Note that codeword bits that are always zero can arise even when H does not have linearly dependent rows. For example, if a row of H has just one 1 in it, the codeword bit at that position must be zero in any codeword. The way the software handles parity check matrices with less than M independent rows is equivalent to adding additional rows to H in which only one bit is 1, in order to produce M independent checks.

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