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372 lines
14 KiB
HTML
372 lines
14 KiB
HTML
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<HTML><HEAD>
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<TITLE> Creating a Parity Check Matrix </TITLE>
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</HEAD><BODY>
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<H1> Creating a Parity Check Matrix </H1>
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<P>This software deals only with linear block codes for binary (ie,
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modulo-2, GF(2)) vectors. The set of valid codewords for a linear
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code can be specified by giving a <I>parity check matrix</I>,
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<B>H</B>, with <I>M</I> rows and <I>N</I> columns. The valid
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codewords are the vectors, <B>x</B>, of length <I>N</I>, for which
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<B>Hx</B>=<B>0</B>, where all arithmetic is done modulo-2. Each row
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of <B>H</B> represents a parity check on a subset of the bits in
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<B>x</B>; all these parity checks must be satisfied for <B>x</B> to be
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a codeword. Note that the parity check matrix for a given code (ie,
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for a given set of valid codewords) is not unique, even after
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eliminating rows of <B>H</B> that are redundant because they are
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linear combinations of other rows.
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<P>This software stores parity check matrices in files in a sparse
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format. These parity-check files are <I>not</I> human-readable
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(except by using the <A HREF="#print-pchk"><TT>print-pchk</TT></A>
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program). However, they <I>are</I> readable on a machine with a
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different architecture than they were written on.
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<P>Some LDPC software by David MacKay and others uses the
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<A HREF="http://www.inference.phy.cam.ac.uk/mackay/codes/alist.html">alist
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format</A> for parity check matrices. Two programs for converting
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between this format and the format for sparse parity check matrices
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used by this software are provided.
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<A NAME="ldpc"><H2>Methods for constructing LDPC codes</H2></A>
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<P>This software is primarily intended for experimentation with Low
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Density Parity Check (LDPC) codes. These codes can be constructed by
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various methods, which generally involve some random selection of
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where to put 1s in a parity check matrix. Any such method for
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constructing LDPC codes will have the property that it produces parity
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check matrices in which the number of 1s in a column is approximately
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the same (perhaps on average) for any size parity check matrix. For a
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given code rate, these matrices therefore become increasingly sparse
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as the length of a codeword, and hence the number of parity checks,
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increases.
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<P>Many methods for constructing LDPC matrices are described in the
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<A HREF="refs.html">references</A>. Two simple methods are currently
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implemented by this software, both of which operate according to the
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following scheme:
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<OL>
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<LI> Create a preliminary parity check matrix by one of the methods.
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<LI> Add 1s to the parity check matrix in order to avoid rows that have no
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1s in them, and hence are redundant, or which have only one 1 in them,
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in which case the corresponding codeword bits will always be zero.
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The places within such a row to add these 1s are selected randomly.
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<LI> If the preliminary parity check matrix constructed in step (1) had
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an even number of 1s in each column, add further 1s to avoid the problem
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that this will cause the rows to add to zero, and hence at least
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one check will be redundant. Up to two 1s are added (since it is also
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undesirable for the sum of the rows to have only one 1 in it), at
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positions selected randomly from the entire matrix. However, the
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number of 1s to add in this step is reduced by the number already added
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in step (2). (Note that although redundant checks are not disastrous,
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they are better avoided; see the discussion of <A HREF="dep-H.html">linear
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dependence in parity check matrices</A>.)
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<LI> If requested, try to eliminate
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situations where a pair of columns both have 1s in a particular pair of
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rows, which correspond to cycles of length four in the factor graph of
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the parity check matrix. When such a situation is detected, one of the
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1s involved is moved randomly within its column. This continues until
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no such situations remain, or until 10 passes over all columns have
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failed to eliminate all such situations.
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</OL>
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<P>The <I>evencol</I> method is the simplest way of performing step
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(1) of the above procedure. For each column of the parity check
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matrix, independently, it places a specified number of 1s in positions
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selected uniformly at random, with the only constraint being that
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these 1s be in distinct rows. Note that despite the name, the columns
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do not have to have the same number of 1s - a distribution over
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several values for the number of 1s in a column can be specified
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instead. Such codes with different-weight columns are sometimes
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better than codes in which every column has the same weight.
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<P>The <I>evenboth</I> method also puts a specified number of 1s in
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each column, but it tries as well to keep the numbers of 1s in the
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rows approximately the same. Initially, it creates indicators for all
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the 1s that will be required, and assigns these 1s to rows as evenly
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as it can, favouring earlier rows if an exactly even split is not
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possible. It then assigns 1s to successive columns by selecting
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randomly, without replacement, from this initial supply of 1s, subject
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only to the constraint that the 1s assigned to a column must be in
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distinct rows. If at some point it is impossible to put the required
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number of 1s in a column by picking from the 1s remaining, a 1 is set
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in that column without reference to other columns, creating a possible
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unevenness.
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<P>Note that regardless of how evenly 1s are distributed in the
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preliminary parity check matrix created in step (1), steps (2) and (3)
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can make the numbers of 1s in the both rows and columns be uneven, and
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step (4), if done, can make the numbers of 1s in rows be uneven.
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<P><A NAME="make-pchk"><HR><B>make-pchk</B>: Make a parity check
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matrix by explicit specification.
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<BLOCKQUOTE><PRE>
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make-pchk <I>pchk-file n-checks n-bits row</I>:<I>col ...</I>
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</PRE></BLOCKQUOTE>
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<P>Creates a file named <TT><I>pchk-file</I></TT> in
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which it stores a parity check matrix with <TT><I>n-checks</I></TT>
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rows and <TT><I>n-bits</I></TT> columns. This parity check matrix
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consists of all 0s except for 1s at the <I>row</I>:<I>col</I>
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positions listed. Rows and columns are numbered starting at zero.
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This program is intended primarily for testing and demonstration
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purposes.
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<P><B>Example:</B> The well-known Hamming code with codewords of
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length <I>N</I>=7 and with <I>M</I>=3 parity checks can be can be
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created as follows:
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<UL><PRE>
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<LI>make-pchk ham7.pchk 3 7 0:0 0:3 0:4 0:5 1:1 1:3 1:4 1:6 2:2 2:4 2:5 2:6
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</PRE></UL>
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<P><A NAME="alist-to-pchk"><HR><B>alist-to-pchk</B>: Convert a parity
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check matrix from alist format to the sparse matrix format used by
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this software.
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<BLOCKQUOTE><PRE>
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alist-to-pchk [ -t ] <I>alist-file pchk-file</I>
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</PRE></BLOCKQUOTE>
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<P>Converts a parity check matrix in
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<A HREF="http://www.inference.phy.cam.ac.uk/mackay/codes/alist.html">alist
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format</A> stored in the file named <TT><I>alist-file</I></TT> to
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the sparse matrix format used by this software, storing it in the
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file named <TT><I>pchk-file</I></TT>.
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<P>If the <B>-t</B> option is given, the transpose of the parity check
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matrix in <TT><I>alist-file</I></TT> is stored in the
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<TT><I>pchk-file</I></TT>.
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<P>Any zeros indexes in the alist file are ignored, so that alist files
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with zero padding (as required in the specification) are accepted,
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but files without this zero padding are also accepted. Newlines
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are ignored by <TT>alist-to-pchk</TT>, so no error is reported if
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the set of indexes in a row or column description are not those
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on a single line.
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<P><A NAME="pchk-to-alist"><HR><B>pchk-to-alist</B>: Convert a parity
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check matrix to alist format.
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<BLOCKQUOTE><PRE>
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pchk-to-alist [ -t ] [ -z ] <I>pchk-file alist-file</I>
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</PRE></BLOCKQUOTE>
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<P>Converts a parity check matrix stored in the sparse matrix format
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used by this software, in the file named <TT><I>pchk-file</I></TT>, to
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the <A
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HREF="http://www.inference.phy.cam.ac.uk/mackay/codes/alist.html">alist
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format</A>, storing it in the file named <TT><I>alist-file</I></TT>.
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<P>If the <B>-t</B> option is given, the transpose of the parity check
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matrix is converted to alist format.
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<P>If the number of 1s is not
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the same for each row or each column, the alist format specification
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says that the list of indexes of 1s for each row or column should
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be padded with zeros to the maximum number of indexes. By default,
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<TT>pchk-to-alist</TT> does this, but output of these 0s can be
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suppressed by specifying the <B>-z</B> option. (The <TT>alist-to-pchk</TT>
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program will accept alist files produced with or without the <B>-z</B>
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option.)
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<P><A NAME="print-pchk"><HR><B>print-pchk</B>: Print a parity check matrix.
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<BLOCKQUOTE><PRE>
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print-pchk [ -d ] [ -t ] <I>pchk-file</I>
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</PRE></BLOCKQUOTE>
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<P>Prints a human-readable representation of the parity check matrix stored
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in <TT><I>pchk-file</I></TT>.
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The <B>-d</B> option causes the matrix to be printed in a dense
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format, even though parity check matrices are always stored in the
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file in a sparse format. If the <B>-t</B> option is present, what is
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printed is the transpose of the parity check matrix.
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<P>The sparse display format consists of one line for every row of the
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matrix, consisting of the row number, a colon, and the column numbers
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at which 1s are located (possibly none). Row and columns numbers
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start at zero. No attempt is made to wrap long lines.
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<P>The dense display is the obvious array of 0s and 1s. Long lines
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are not wrapped.
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<P><B>Example</B>: The parity check matrix for the Hamming code created
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by the example for <A HREF="#make-pchk"><TT>make-pchk</TT></A> would print
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as follows:
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<UL><PRE>
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<LI>print-pchk ham7.pchk
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Parity check matrix in ham7.pchk (sparse format):
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0: 0 3 4 5
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1: 1 3 4 6
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2: 2 4 5 6
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<LI>print-pchk -d ham7.pchk
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Parity check matrix in ham7.pchk (dense format):
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1 0 0 1 1 1 0
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0 1 0 1 1 0 1
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0 0 1 0 1 1 1
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</PRE></UL>
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<P><A NAME="make-ldpc"><HR><B>make-ldpc</B>: Make a low density parity
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check matrix, by random generation.
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<BLOCKQUOTE><PRE>
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make-ldpc <I>pchk-file n-checks n-bits seed method</I>
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</PRE>
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<BLOCKQUOTE>
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where <TT><I>method</I></TT> is one of the following:
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<BLOCKQUOTE><PRE>
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evencol <I>checks-per-col</I> [ no4cycle ]
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evencol <I>checks-distribution</I> [ no4cycle ]
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evenboth <I>checks-per-col</I> [ no4cycle ]
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evenboth <I>checks-distribution</I> [ no4cycle ]
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</PRE></BLOCKQUOTE>
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</BLOCKQUOTE>
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</BLOCKQUOTE>
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<P>Creates a Low Density Parity Check matrix with
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<TT><I>n-checks</I></TT> rows and <TT><I>n-bits</I></TT> columns. The
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parity check matrix will be generated pseudo-randomly by the indicated
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method, using a pseudo-random number stream determined by <TT><I>seed</I></TT>.
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The actual random number seed used is 10 times <TT><I>seed</I></TT> plus 1,
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so as to avoid using the same stream as any of the other programs.
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<P>Two methods are currently available for creating the LDPC matrix,
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specified by <TT>evencol</TT> or <TT>evenboth</TT>. Both methods
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produce a matrix in which the number of 1s in each column is
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approximately <TT><I>checks-per-col</I></TT>, or varies from column
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to column according the the <TT><I>checks-distribution</I></TT>.
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The <TT>evenboth</TT> method also tries to make the number of checks per row be
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approximately uniform; if this is not achieved, a message saying that
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how many bits were placed unevenly is displayed on standard error.
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<P>For both methods, the <TT>no4cycle</TT> option will cause cycles of
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length four in the factor graph representation of the code to be
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eliminated (if possible). A message is displayed on standard error if
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this is not achieved.
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<P>A <TT><I>checks-distribution</I></TT> has the form
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<BLOCKQUOTE><PRE>
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<I>prop</I>x<I>count</I>/<I>prop</I>x<I>count</I>/...
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</PRE></BLOCKQUOTE>
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Here, <TT><I>prop</I></TT> is a proportion of columns that have the
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associated <TT><I>count</I></TT>. The proportions need not sum to one,
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since they will be automatically normalized. For example, <TT>0.3x4/0.2x5</TT>
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specifies that 60% of the columns will contain four 1s and 40% will
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contain five 1s.
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<P>See the <A HREF="#ldpc">discussion above</A> for more details
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on how these methods construct LDPC matrices.
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<P><B>Example 1:</B> The <TT>make-ldpc</TT> command below creates
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a 20 by 40 low density parity check matrix with three 1s per
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column and six 1s per row, using random seed 1. The matrix
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is then printed in sparse format
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using <A HREF="#print-pchk">print-pchk</A>.
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<UL><PRE>
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<LI>make-ldpc ldpc.pchk 20 40 1 evenboth 3
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<LI>print-pchk ldpc.pchk
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Parity check matrix in ldpc.pchk (sparse format):
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0: 10 14 18 27 38 39
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1: 2 3 5 11 27 30
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2: 15 19 20 21 24 26
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3: 2 4 25 28 32 38
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4: 7 9 12 22 33 34
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5: 5 6 21 22 26 32
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6: 1 4 13 24 25 28
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7: 1 14 28 29 30 36
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8: 11 13 22 23 32 37
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9: 6 8 13 20 31 33
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10: 0 3 24 29 31 38
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11: 7 12 15 16 17 23
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12: 3 16 29 34 35 39
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13: 0 8 10 18 36 37
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14: 6 11 18 20 35 39
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15: 0 7 14 16 25 37
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16: 2 4 9 19 30 31
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17: 5 9 10 17 19 23
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18: 8 15 17 21 26 27
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19: 1 12 33 34 35 36
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</PRE></UL>
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<P><B>Example 2:</B> The two <TT>make-ldpc</TT> commands
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below both create a 20 by 40 low density parity check matrix with 30%
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of columns with two 1s, 60% of columns with three 1s, and 10% of
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columns with seven 1s. The transpose of the parity check matrix
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is then printed in sparse format.
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<UL><PRE>
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<LI>make-ldpc ldpc.pchk 20 40 1 evenboth 0.3x2/0.6x3/0.1x7
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<LI>make-ldpc ldpc.pchk 20 40 1 evenboth 3x2/6x3/1x7
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<LI>print-pchk -t ldpc.pchk
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Transpose of parity check matrix in ldpc.pchk (sparse format):
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0: 13 16
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1: 9 18
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2: 1 10
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3: 3 15
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4: 4 14
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5: 14 17
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6: 4 5
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7: 1 8
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8: 0 4
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9: 9 14
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10: 5 8
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11: 6 16
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12: 2 12 19
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13: 3 17 18
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14: 2 16 17
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15: 2 11 18
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16: 12 13 19
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17: 7 13 18
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18: 2 5 11
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19: 10 12 14
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20: 1 8 16
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21: 10 18 19
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22: 3 6 17
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23: 7 11 12
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24: 1 2 19
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25: 0 6 7
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26: 5 8 15
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27: 1 4 7
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28: 6 13 19
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29: 3 4 11
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30: 3 8 17
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31: 4 5 9
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32: 0 10 15
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33: 7 11 13
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34: 8 12 19
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35: 0 2 10
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36: 0 5 9 11 15 17 18
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37: 0 1 2 6 7 14 16
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38: 0 1 3 9 12 13 15
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39: 3 6 9 10 14 15 16
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</PRE></UL>
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