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86 lines
3.0 KiB
HTML
86 lines
3.0 KiB
HTML
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<HTML><HEAD>
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<TITLE> Linear Dependence in Parity Check Matrices </TITLE>
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</HEAD><BODY>
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<H1> Linear Dependence in Parity Check Matrices </H1>
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<P>If a code is specified by means of a <I>M</I> by <I>N</I> parity
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check matrix, <B>H</B>, in which some rows are linearly dependent - a
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situation that is usually avoided - it would be possible to map more
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than the usual <I>K=N-M</I> message bits into a codeword, since one or
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more rows of <B>H</B> could have been deleted without affecting which
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bit vectors are codewords.
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<P>However, this software does not increase the number of message bits
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in this case, but instead produces a generator matrix in which some
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rows are all zero, which will cause some bits of the codeword to
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always be zero, regardless of the source message. Referring to the <A
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HREF="encoding.html#gen-rep">description of generator matrix
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representations</A>, this is accomplished by partially computing
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what would normally be <B>A</B><SUP><SMALL>-1</SMALL></SUP> (for a
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dense or mixed representations) or the <B>L</B> and <B>U</B> matrices
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(for a sparse representation), even though singularity prevents this
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computation from being carried out fully.
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<P><B>Example:</B> The parity check matrix created below is redundant,
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since the 10100 row is equal to the sum of the 11000 and 01100 rows.
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<UL><PRE>
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<LI>make-pchk dep.pchk 4 5 0:0 0:1 1:1 1:2 2:0 2:2 3:3 3:4
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<LI>print-pchk -d dep.pchk
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Parity check matrix in dep.pchk (dense format):
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1 1 0 0 0
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0 1 1 0 0
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1 0 1 0 0
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0 0 0 1 1
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<LI>make-gen dep.pchk dep.gen dense
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Note: Parity check matrix has 1 redundant checks
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Number of 1s per check in Inv(A) X B is 0.2
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<LI>print-gen dep.gen
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Generator matrix (dense representation):
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Column order (message bits at end):
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0 1 3 2 4
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Inv(A) X B:
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0
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0
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1
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0
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</PRE></UL>
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The generator matrix above can be used to encode message blocks containing
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one bit. This message bit is copied unchanged to the last bit (numbered 4)
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of the codeword, and the first four bits of the codeword are set by multiplying
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this message bit (seen as a vector of length one) by
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<B>A</B><SUP><SMALL>-1</SMALL></SUP><B>B</B>, shown above, and then
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storing the results in positions given by the column ordering.
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The result is that bit 3 of the codeword produced is
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also set to the message bit, and bits 0, 1, and 2 are set to zero.
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<P>Which bits are used for message bits, and which bits are fixed at
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zero, depends on arbitrary choices in the algorithm, which may differ
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from one encoding method to another. No attempt is made to make the
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best choice.
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<P>Note that codeword bits that are always zero can arise even when <B>H</B>
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does not have linearly dependent rows. For example, if a row of <B>H</B>
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has just one 1 in it, the codeword bit at that position must be zero in any
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codeword. The way the software handles parity check matrices with less
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than <I>M</I> independent rows is equivalent to adding additional rows
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to <B>H</B> in which only one bit is 1, in order to produce <I>M</I>
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independent checks.
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<HR>
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<A HREF="index.html">Back to index for LDPC software</A>
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</BODY></HTML>
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