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On the Design of Fast Convergent LDPC Codes for the BEC: An Optimization Approach

机译：BEC的快速收敛LDPC码设计：一种优化方法

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摘要

The complexity-performance trade-off is a fundamental aspect of the design of low-density parity-check (LDPC) codes. In this paper, we consider LDPC codes for the binary erasure channel (BEC), use code rate for performance metric, and number of decoding iterations to achieve a certain residual erasure probability for complexity metric. We first propose a quite accurate approximation of the number of iterations for the BEC. Moreover, a simple but efficient utility function corresponding to the number of iterations is developed. Using the aforementioned approximation and the utility function, two optimization problems w.r.t. complexity are formulated to find the code degree distributions. We show that both optimization problems are convex. In particular, the problem with the proposed approximation belongs to the class of semi-infinite problems which are computationally challenging to be solved. However, the problem with the proposed utility function falls into the class of semi-definite programming (SDP) and thus, the global solution can be found efficiently using available SDP solvers. Numerical results reveal the superiority of the proposed code design compared to existing code designs from literature.
机译：复杂度与性能的折衷是低密度奇偶校验（LDPC）码设计的基本方面。在本文中，我们考虑将LDPC码用于二进制擦除信道（BEC），将码率用于性能指标，并使用解码迭代次数为复杂性指标获得一定的残余擦除概率。我们首先提出一个非常精确的BEC迭代次数的近似值。此外，开发了与迭代次数相对应的简单但有效的效用函数。使用上述近似值和效用函数，有两个优化问题。制定复杂度以找到代码度分布。我们证明两个优化问题都是凸的。特别地，所提出的近似的问题属于半无限问题的类别，其在计算上难以解决。但是，所提出的效用函数的问题属于半定规划（SDP）类别，因此，可以使用可用的SDP求解器有效地找到全局解。数值结果表明，与文献中现有的代码设计相比，该代码设计具有优越性。

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