Free distance bounds for protograph-based regular LDPC convolutional codes

Abstract

In this paper asymptotic methods are used to form lower bounds on the free distance to constraint length ratio of several ensembles of regular, asymptotically good, protographbased LDPC convolutional codes. In particular, we show that the free distance to constraint length ratio of the regular LDPC convolutional codes exceeds that of the minimum distance to block length ratio of the corresponding LDPC block codes.

Asymptotic trapping set analysis of regular protograph-based LDPC convolutional code ensembles

“…7. Just as we expect the tail-biting ensemble zero-contour curve values δ (λ) ts (0) to correspond to minimum distance growth rates of the associated block code ensembles, we expect that the lower bound on the free distance growth rate δ f ree of the regular convolutional codes (found in [9]) based on the displayed cuts correspond to the value δ ccts (0) of the convolutional lower bound zero-contour curve. This can be seen in Figure 8, where the lower bounds on δ ccts (∆) = α are plotted against ∆α = β to form a convolutional lower bound zero-contour curve for each of the regular convolutional ensembles considered.…”

“…As described in Section V, we make use of ensembles of tail-biting LDPC convolutional codes to obtain a lower bound on the desired ∆-trapping set growth rate of the associated unterminated convolutional code family. 5 If the syndrome former matrix is not in minimal form, (9) results in an upper bound on νe, which implies that δccts(∆) is underestimated in this case.…”

“…The protograph is then unwrapped according to the nonuniform cutting vector ξ method as described in Section III-A. The nonuniform cuts chosen are consistent with those previously presented in [9], and they are displayed in Fig. 7.…”

“…Further, in [8] and [9], it was shown that several ensembles of both regular and irregular unterminated periodically time-varying LDPC convolutional codes based on protographs are asymptotically good. In other words, their average free distance grows linearly with constraint length.…”

“…The analysis used in [8] and [9] to calculate ensemble average weight enumerators can be extended to the problem of finding ensemble average trapping set enumerators. In this paper, building on work by Abu-Surra, Ryan, and Divsalar [10], asymptotic methods are used to calculate a lower bound on the average trapping set enumerators for several ensembles of regular, asymptotically good, protograph-based LDPC convolutional codes.…”