ITW 1999. Metsovo. Greece, June 27 - July 1
`Constructing Low-Density Parity-Check Codes with Circulant Matrices
`J. W. Bond, S. Huit, and H. Schmidt4
`Science Applications International Corp, 4105 Hancock St, San Diego, CA, 92110, e-mail: bond-jwOmarlin.nosc.mil
`-t Department of Mathematics, San Diego State University, San Diego, CA 92182, e-mail: huiOsaturn.sdsu.edu
`3 Technology Service Corporation, 962 Wayne Ave, Suite 800, Silver Spring, MD 20910, e-mail: hschmidtOtscwo.com
`a0 + a12 + + an-1zn-l is primitive in Fz[x], then
`each row of the inverse of C has asymptotically the same
`number of zeros and ones in each row. Using this fact on
`the inverse of primitive circulant matrices, it is not hard to
`see that the average weight of the check-bits is about m/2
`(see Bond [2]), where n is the number of check-bits. This
`guarantees good performance on the average when circulant
`matrices corresponding to primitive polynomials are used in
`the construction.
`111. RANDOM AND CIRCULANT CONSTRUCTION WITH
`CONSTRAINTS
`By using a circulant matrix that corresponds to a primitive
`polynomial in the construction of the parity-check matrix, we
`can guarantee that the average weight of the codewords is at
`least m/2. However, it is possible to have low weight code-
`words. We attempt to eliminate the low weight codewords by
`carefully choosing the random part of the parity-check matrix.
`The codewords are vectors of the form
`
`Abstract - This is a report on our ongoing effort
`to implement Low-Density Parity-Check codes with
`Iterative Belief Propagation decoding in a communi-
`cation system. The system requires the codes to have
`block lengths from 1000 to 2000 bits. We describe two
`different methods of constructing the codes using: (1)
`a combination of random and circulant matrices, and
`(2) random and circulant matrices with constraints
`to control the number of low weight codewords. We
`illustrate the performances of the different construc-
`tions with simulations.
`
`I. INTRODUCTION
`Sparse matrix parity-check code was introduced by Gal-
`lager [4] and has attracted much attention recently; see, for
`example, MacKay [6], Luby et al[5], and Bond [l].
`The most common method of implementing the sparse ma-
`trix method is to randomly construct a m x (n + m) parity-
`check matrix
`H = [ R C ] ,
`where R is m x n and C is m xm with certain desired properties
`and then put it into systematic form
`[ C-lR I ] .
`
`The most common condition to impose on the parity-check
`matrix H is that the locations of 1’s of any two different rows
`be different with at most one exception. The generator matrix
`m i
`is then
`C-’R 1.
`
`where b E F;. For w to be a low weight codeword, b and
`a = C-’Rb must have low weights. So to have low weight
`codewords, the equation Rb = Ca must be solvable by low
`weight vectors b, a. We choose R carefully so that Rh = Ca
`has few or no solutions with low weight a’s and b’s.
`IV. CONCLUSIONS
`Low-density parity-check codes with excellent average dis-
`tance properties can be constructed by using parity-check ma-
`trices that are concatenations of circulant matrices and ran-
`dom matrices. However, these codes may have low minimum
`weights and this can give block error rates that are unaccept-
`able for many important applications. In this talk we present
`the block error rate for codes constructed using circulant ma-
`trices and codes that are obtained by carefully choosing the
`circulant matrix and random matrix to ensure that the result-
`ing codes have few or no low weight codewords.
`
`REFERENCES
`[l] Bond, J. W., Hui, S. and Schmidt, H. (1997) Low density par-
`ity check codes based on sparse matrices with no small cycles,
`Proceedings of the IMA.
`[2] Bond, J. W., Hui, S. and Schmidt, H. (1999) Sparse matrix
`parity-check codes and circulant matrices, In preparation.
`[3] Bond, J. W., Hui, S. and Schmidt, H. (1999) The Euclidean al-
`gorithm and primitive polynomials over finite fields, submitted.
`[4] Gallager, R. G. (1963) Low Density Parity-Check Codes, MIT
`Press.
`[5] Luby, M., Mitzenmacher, M., Shokrollahi, M. A., Spielman, D.
`(1998) Analysis of Low Density Codes and Improved Designs
`using Irregular Graphs, Preprint.
`[6] MacKay, D. J. C. (1998) Good error-correcting codes based on
`very sparse matrices, Preprint.
`
`52
`
`Of course, there is no guarantee that the matrix C is invert-
`ible. Indeed, it is quite likely for C to be non-invertible. Note
`that C is not invertible in F z if its base 10 determinant is even.
`11. CONSTRUCTION USING CIRCULANT MATRICES
`In our implementation, we use a circulant matrix C. Recall
`that a square matrix is circulant if each row of the matrix is
`the cyclic shift of the previous row. An example of a circulant
`matrix is
`
`[H i
`
`1
`
`1
`
`y]
`
`0
`
`0
`
`
`
`It is easy to see that the inverse of a circulant matrix is again
`circulant. The use of a circulant matrix allows us to guaran-
`tee invertibility and to control the number of cycles. Another
`advantage of a circulant matrix is that it is automatically reg-
`ular. The most important reason for using a circulant matrix
`is the following mathematical fact.
`Let C be a circulant matrix with
`[ a0 a1
`
`a,-1 3. We showed in Bond [3] that if
`
`first
`
`row
`
`Hughes, Exh. 1030, p. 1
`
`