Low-density parity-check (LDPC) codes are among the most prominent
error-correction schemes. They find application to fortify various modern
storage, communication, and computing systems. Protograph-based (PB) LDPC codes
offer many degrees of freedom in the code design and enable fast encoding and
decoding. In particular, spatially-coupled (SC) and multi-dimensional (MD)
circulant-based codes are PB-LDPC codes with excellent performance. Efficient
finite-length (FL) algorithms are required in order to effectively exploit the
available degrees of freedom offered by SC partitioning, lifting, and MD
relocations. In this paper, we propose a novel Markov chain Monte Carlo (MCMC
or MC$^2$) method to perform this FL optimization, addressing the removal of
short cycles. While iterating, we draw samples from a defined distribution
where the probability decreases as the number of short cycles from the previous
iteration increases. We analyze our MC$^2$ method theoretically as we prove the
invariance of the Markov chain where each state represents a possible
partitioning or lifting arrangement. Via our simulations, we then fit the
distribution of the number of cycles resulting from a given arrangement on a
Gaussian distribution. We derive estimates for cycle counts that are close to
the actual counts. Furthermore, we derive the order of the expected number of
iterations required by our approach to reach a local minimum as well as the
size of the Markov chain recurrent class. Our approach is compatible with code
design techniques based on gradient-descent. Numerical results show that our
MC$^2$ method generates SC codes with remarkably less number of short cycles
compared with the current state-of-the-art. Moreover, to reach the same number
of cycles, our method requires orders of magnitude less overall time compared
with the available literature methods.

Este artículo explora los viajes en el tiempo y sus implicaciones.

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