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<HTML><HEAD><TITLE> Linear Dependence in Parity Check Matrices </TITLE></HEAD><BODY><H1> Linear Dependence in Parity Check Matrices </H1><P>If a code is specified by means of a <I>M</I> by <I>N</I> paritycheck matrix, <B>H</B>, in which some rows are linearly dependent - asituation that is usually avoided - it would be possible to map morethan the usual <I>K=N-M</I> message bits into a codeword, since one ormore rows of <B>H</B> could have been deleted without affecting whichbit vectors are codewords.<P>However, this software does not increase the number of message bitsin this case, but instead produces a generator matrix in which somerows are all zero, which will cause some bits of the codeword toalways be zero, regardless of the source message. Referring to the <AHREF="encoding.html#gen-rep">description of generator matrixrepresentations</A>, this is accomplished by partially computingwhat would normally be <B>A</B><SUP><SMALL>-1</SMALL></SUP> (for adense or mixed representations) or the <B>L</B> and <B>U</B> matrices(for a sparse representation), even though singularity prevents thiscomputation from being carried out fully.<P><B>Example:</B> The parity check matrix created below is redundant,since the 10100 row is equal to the sum of the 11000 and 01100 rows.<UL><PRE><LI>make-pchk dep.pchk 4 5 0:0 0:1 1:1 1:2 2:0 2:2 3:3 3:4<LI>print-pchk -d dep.pchkParity 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<LI>make-gen dep.pchk dep.gen denseNote: Parity check matrix has 1 redundant checksNumber of 1s per check in Inv(A) X B is 0.2<LI>print-gen dep.genGenerator matrix (dense representation):Column order (message bits at end): 0 1 3 2 4Inv(A) X B: 0 0 1 0</PRE></UL>The generator matrix above can be used to encode message blocks containingone 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 multiplyingthis message bit (seen as a vector of length one) by <B>A</B><SUP><SMALL>-1</SMALL></SUP><B>B</B>, shown above, and thenstoring the results in positions given by the column ordering.The result is that bit 3 of the codeword produced isalso set to the message bit, and bits 0, 1, and 2 are set to zero.<P>Which bits are used for message bits, and which bits are fixed atzero, depends on arbitrary choices in the algorithm, which may differfrom one encoding method to another. No attempt is made to make thebest choice.<P>Note that codeword bits that are always zero can arise even when <B>H</B>does not have linearly dependent rows. For example, if a row of <B>H</B>has just one 1 in it, the codeword bit at that position must be zero in anycodeword. The way the software handles parity check matrices with lessthan <I>M</I> independent rows is equivalent to adding additional rowsto <B>H</B> in which only one bit is 1, in order to produce <I>M</I> independent checks.<HR><A HREF="index.html">Back to index for LDPC software</A></BODY></HTML>⌨️ 快捷键说明
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