Error-Trellis State Complexity of LDPC Convolutional Codes Based on Circulant Matrices

TL;DR

This paper investigates how cyclic row shifts in LDPC convolutional codes derived from QC codes can reduce error-trellis state complexity while maintaining error correction performance.

Contribution

It demonstrates that applying controlled row shifts to the parity-check matrix can significantly lower the error-trellis state complexity without compromising the code's error-correcting ability.

Findings

State-space complexity can be reduced through row shifts.

Free distance is preserved or lower-bounded after shifting.

Controlled shifts optimize trellis complexity.

Abstract

Let H(D) be the parity-check matrix of an LDPC convolutional code corresponding to the parity-check matrix H of a QC code obtained using the method of Tanner et al. We see that the entries in H(D) are all monomials and several rows (columns) have monomial factors. Let us cyclically shift the rows of H. Then the parity-check matrix H'(D) corresponding to the modified matrix H' defines another convolutional code. However, its free distance is lower-bounded by the minimum distance of the original QC code. Also, each row (column) of H'(D) has a factor different from the one in H(D). We show that the state-space complexity of the error-trellis associated with H'(D) can be significantly reduced by controlling the row shifts applied to H with the error-correction capability being preserved.