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.

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.

Back to index for LDPC software