Optimizing Code Parameters of Finite-Length SC-LDPC Codes Using the Scaling Law

In this paper, we optimize code parameters of finite-length spatially coupled low-density parity-check (SC-LDPC) codes, represented by a set of code parameters $(l,r,w,L,M)$ . Although the finite-length scaling behavior of SC-LDPC codes was studied in the existing literature, the previous works impose a constraint such that the coupling width $w$ is equal to the variable node degree $l$ and they do not focus on optimizing the code parameters for given code and decoder specifications such as the code rate, frame size, and decoding complexity. In order to optimize the code parameters with the target specifications, we first extend the scaling law of SC-LDPC codes without the constraint $w=l$ . Using the scaling law formulated with a new variable $w$ , we show that the coupling width $w$ directly affects the slope of the performance curve and performance comparisons are given to investigate trade-offs inherent in the code parameters. It is shown that there are trade-offs for the code parameters in the perspective of the asymptotic performance limit, code rate, and scaling behaviors. In addition, the scaling law allows us to find the optimal code parameter set showing the best finite-length performance. Interestingly, the optimal code parameter set $(l,r,w)$ varies depending on the coupling length $L$ and uncoupled code length $M$ that determine the code and decoder specifications, which means there is no specific code parameter set prevailing over different kinds of applications. Finally, we illustrate this result using the investigated trade-offs on the code parameters, which gives us useful insight on how to choose the code parameters.